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Problem 701

e the first term is 9 and the third term is $\frac{1}{81}$ . (Why are there two possible answers?) Find the 4 th term in the geometric sequence where the first term is 6 and the 7 th term is $\frac{3}{32}$ .
For the geometric sequence $3, m, n, 192, \ldots$ , find the values for $m$ and $n$ . Find the value of $x$ such that the following sequence forms a geometric progression: $x-1,3 x+4,6 x+8$

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Problem 702

Use the arithmetic sequence formula to find the 38th number in the sequence: $3,6,9 \ldots$

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Problem 703

16 / Final (April 25, 2024) - [9 points] The Taylor series centered at $x=1$ for a function $T(x)$ is given by: $T(x)=\sum_{n=0}^{\infty} \frac{(n!)^{2}}{(-5)^{n} \cdot(2 n)!}(x-1)^{4 n+3}$ a. [6 points] Find the radius of convergence of the Taylor series above. Show your work. Do not attempt to find the interval of convergence. page 1 x - 9 points] The Taylor series ces $T(x)=\sum_{n=0}^{\infty} \frac{(n!)^{2}}{(-5)^{n} \cdot(2 n)!}(x-1)^{4 n+3} \text { is given by: }$ - $\begin{array}{l} \frac{\left((n+1) \frac{\sqrt{k}}{\varepsilon}\right)^{2}}{(-5)^{\frac{2+2}{2}}} \frac{(2 n+2))}{(-5)} \\ 1^{(x} \\ \text { - } 1)^{4 n+7} \\ \frac{(25)^{2} \cdot(2 n)^{1}}{\left(x^{2}\right)^{\alpha}(x-1)^{4 n+3}} \\ \frac{(2 n+2)(2 n+1)(2 n)}{-5(2 n+1)(n+1)} \\ 2 n! \\ (n+2 n+2 \\ \left.(n+1)^{\prime}\right)^{2(n+1)} \end{array}$ $\begin{array}{c} 2 \\ 6+2=8 \\ 2+2=4 \\ 2+2+6 \end{array}$ $-60_{4} \cdot$ 1 b. [3 points] Compute $T^{(123)}(1)$ . Show your work. You do not need to simplify your answer. Answer: $\quad T^{(123)}(1)=$ $\qquad$

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Problem 704

2) Determine how much of the total loan payment applies toward principal and how much applies toward interest for a student loan of \$38,156 at a fixed APR of 8\% for 11 years. A) \$38,156 pays off the principal and \$19,321.94 represents interest payments. B) \$38,156 pays off the principal and \$19,398.46 represents interest payments. C) \$38,156 pays off the principal and \$19,362.76 represents interest payments. D) \$38,156 pays off the principal and \$19,338.95 represents interest payments.

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Problem 705

Type the missing number in this sequer 1, 3, 9, 27, 81,

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Problem 706

Question 3 (1 point) Imagine you need $\$ 425$ to cover unexpected car repairs. You decide to get a payday loan. The payday lender charges a $\$ 30$ fee for a two-week loan. If this loan was to be rolled over for an entire year then how much would be owed?
Your Answer: $\square$ Answer

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Problem 707

[10] 3. For $f(x)=-32 x+75$ and $g(x)=7(0.6)^{x}$ find the following by only using a proper formula: (A) $\sum_{k=0}^{69} f(k)=f(0)+f(1)+f(2)+\cdots+f(69)$ . (B) $\sum_{h=0}^{44} g(h)=g(0)+g(1)+g(2)+\cdots+g(44)$ . (C) What is $\sum_{h=0}^{\infty} g(h)=g(0)+g(1)+g(2)+\cdots+g(n)+g(n+1)+\ldots$ ?

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Problem 708

An investor deposits \$1200 yearly in an annuity at 7% interest. Find the total value after 1, 2, and 3 years.

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Problem 709

Keiko deposits \ $950 annually in a 6% annuity. Find values at the end of years 1, 2, and 3: (a)$ V_1 $, (b)$ V_2 $, (c)$ V_3$.

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Problem 710

Ravi deposits \$625 yearly in an annuity at 8% interest. Find its value at the end of years 1, 2, and 3.

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Problem 711

Write a function for the arithmetic sequence of Kaia's savings: $\$ 525$ , $\$ 580$ , $\$ 635$ , $\$ 690$ .

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Problem 712

Explain the difference between a sequence and a series. A sequence is a function whose domain is the set of positive integers. A series is a summation of the terms of a sequence. A sequence is a summation whose domain is the set of positive integers. A series is a function of the terms of a sequence. A series is a summation whose domain is the set of positive integers. A sequence is a function of the terms of a series. A series is a function whose domain is the set of positive integers. A sequence is a summation of the terms of a series.

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Problem 713

$\text{Write a formula for the general term (the nth term) of the arithmetic sequence } 5, 2, -1, -4. \text{ Then use the formula for } a_n \text{ to find } a_{20}.$

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Problem 714

Доказательство. Мы воспользуемся леммой 2.1.10 в случае необходимости и тогда можем считать, что неравенства выполнены для всех $n \geq 0$ . (1) Имеем для каждого $n \geq 2$ $x_{n}=x_{1} \cdot \frac{x_{2}}{x_{1}} \cdot \frac{x_{3}}{x_{2}} \cdots \frac{x_{n-1}}{x_{n-2}} \cdot \frac{x_{n}}{x_{n-1}}$
по условию $\frac{x_{2}}{x_{1}}, \frac{x_{3}}{x_{2}}, \ldots, \frac{x_{n}}{x_{n-1}} \leq q<1$
тогда $x_{n}=x_{1} \cdot \frac{x_{2}}{x_{1}} \cdot \frac{x_{3}}{x_{2}} \cdots \frac{x_{n-1}}{x_{n-2}} \cdot \frac{x_{n}}{x_{n-1}} \leq x_{1} q^{n-1}$
С другой стороны, (см. пример 2.1.4) ряд ( $q^{n}$ ) сходится при $q<1$ , а тогда по предложению 2.1.8 ряд ( $x_{1} q^{n}$ ) тоже сходится. Наконец, по признаку 1 (Следствие 2.2.2) ряд $\left(x_{n}\right)$ сходится.

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Problem 715

\begin{problem} Докажите признаки Даламбера для сходимости и расходимости ряда.
1. Если для почти всех $n \geq 0$ $\frac{x_{n+1}}{x_{n}} \leq q < 1,$ то ряд $\left(x_{n}\right)$ сходится; если же для почти всех $n \geq 0$ $\frac{x_{n+1}}{x_{n}} \geq 1,$ то ряд $\left(x_{n}\right)$ расходится.
2. Если $\lim_{n \rightarrow \infty} \frac{x_{n+1}}{x_{n}} = q,$ то ряд $\left(x_{n}\right)$ при $q < 1$ сходится, а при $1 < q \leq \infty$ расходится.
\textbf{Доказательство:} Мы воспользуемся леммой 2.1.10 в случае необходимости и тогда можем считать, что неравенства выполнены для всех $n \geq 0$ .
(1) Имеем для каждого $n \geq 2$ $x_{n} = x_{1} \cdot \frac{x_{2}}{x_{1}} \cdot \frac{x_{3}}{x_{2}} \cdots \frac{x_{n-1}}{x_{n-2}} \cdot \frac{x_{n}}{x_{n-1}}$ по условию $\frac{x_{2}}{x_{1}}, \frac{x_{3}}{x_{2}}, \ldots, \frac{x_{n}}{x_{n-1}} \leq q < 1,$ тогда $x_{n} = x_{1} \cdot \frac{x_{2}}{x_{1}} \cdot \frac{x_{3}}{x_{2}} \ldots \frac{x_{n-1}}{x_{n-2}} \cdot \frac{x_{n}}{x_{n-1}} \leq x_{1} q^{n-1}.$
С другой стороны, (см. пример 2.1.4) ряд $(q^{n})$ сходится при $q < 1$ , а тогда по предложению 2.1.8 ряд $(x_{1} q^{n})$ тоже сходится. Наконец, по признаку 1 (Следствие 2.2.2) ряд $(x_{n})$ сходится.
(2) Если же $\frac{x_{2}}{x_{1}}, \frac{x_{3}}{x_{2}}, \ldots, \frac{x_{n}}{x_{n-1}} > 1,$ то $x_{n} \geq x_{1}, \quad n \geq 2.$
Ряд $\left(y_{n}\right)$ , где все $y_{n} = x_{1}$ , очевидно, расходится, тогда по признаку 1 (Следствие 2.2.2), ряд $\left(x_{n}\right)$ тоже расходится.
(3) Пусть $\lim_{n \rightarrow \infty} \frac{x_{n+1}}{x_{n}} = q$ , тогда по определению 1.2.5 для любого $\varepsilon > 0$ существует такой $N$ , что для всех $n \geq N$ имеют место неравенства $q - \varepsilon < \frac{x_{n+1}}{x_{n}} < q + \varepsilon.$
Если $q < 1$ , то пусть $q + \varepsilon < 1$ , тогда для всех $n \geq N, \frac{x_{n+1}}{x_{n}} \leq 1$ , тогда по доказанному признаку (1) ряд $(x_{n})$ сходится.
Если $q > 1$ , то возьмём $\varepsilon > 0$ такое, что $q - \varepsilon > 1$ . Но $\frac{x_{n+1}}{x_{n}} > q - \varepsilon$ при каких-то $n \geq N$ , поэтому для $N \leq n_{0} < n$ получаем $x_{n} = \frac{x_{n}}{x_{n-1}} \cdot \frac{x_{n-1}}{x_{n-2}} \cdots \frac{x_{n_{0}+1}}{x_{n_{0}}} x_{n_{0}} > (q - \varepsilon)^{n-n_{0}} x_{n_{0}}.$ \end{problem}

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Problem 716

2.2.3 Инвариантность суммы
Докажем инвариантность суммы сходящегося положительного ряда при произвольной перестановки его элементов.
Теорема 2.2.10. Пусть $\left(x_{n}\right)$ - сходящийся положительный ряд с суммой S. Тогда полученный в результате произвольной перестановки его элементов новый (заново перенумерованный) ряд также сходится и имеет ту же сумму s. 38
Доказательство. Пусть $x_{1}^{\prime}=x_{n_{1}}, \ldots, x_{k}^{\prime}=x_{n_{k}}, \ldots$ , и пусть $n:=\max \left\{n_{1}, \ldots, n_{k}\right\}$ , рассмотрим тогда частичные суммы $\mathrm{S}_{n}:=x_{1}+x \cdots+x_{n}, \quad \mathrm{~S}_{k}^{\prime}:=x_{1}^{\prime}+\cdots+x_{k}^{\prime},$
так как $1 \leq n_{1}, \ldots, n_{k} \leq n$ и $\left(x_{n}\right)$ - положительный ряд, то $S_{k}^{\prime} \leq S_{n}$
Но, положительный ряд ( $x_{n}$ ) сходится, а тогда по критерию сходимости положительного ряда (см. Теорема 2.2.1), последовательность $\left(\mathrm{S}_{n}\right)$ ограничена, и более того $\mathrm{S}_{n} \leq \mathrm{S}$ для всех $n$ . Таким образом, для всех $k$ получаем $\mathrm{S}_{k}^{\prime} \leq \mathrm{S}_{n} \leq \mathrm{S},$ m.e., последовательность $\left(S_{k}^{\prime}\right)$ частичных сумм ряда $\left(x_{k}^{\prime}\right)$ ограничена, а тогда по критерию

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Problem 717

Доказательство. Воспользовавшись леммой 2.1.10, мы можем считать, что $\left|a_{n}\right| \geq\left|a_{n+1}\right|$ для всех $n$ . Для удобства положим, что первый элемент ряда - это $a_{0}$ , m.e. $n \geq 0$ . Рассмотрим частичную сумму $S_{2 n+1}$ , имеем $\begin{aligned} \mathrm{S}_{2 n+1} & =\left|a_{0}\right|-\left|a_{1}\right|+\left|a_{2}\right|-\left|a_{3}\right|+\left|a_{4}\right|+\cdots+\left|a_{2 n}\right|-\left|a_{2 n+1}\right| \\ & =\left|a_{0}\right|-\left(\left|a_{1}\right|-\left|a_{2}\right|\right)-\left(\left|a_{3}\right|-\left|a_{4}\right|\right)-\cdots-\left(\left|a_{2 n-1}\right|-\left|a_{2 n}\right|\right)-\left|a_{2 n+1}\right| \end{aligned}$
так как $\left|a_{n}\right| \geq\left|a_{n+1}\right|$ , то каждая скобка положительна, это значит, что $S_{2 n+1} \leq\left|a_{0}\right|$ , т.е. последовательность ( $\mathrm{S}_{2 n+1}$ ) ограничена сверху.
С другой стороны, мы можем записать $\begin{aligned} \mathrm{S}_{2 n+1} & =\left|a_{0}\right|-\left|a_{1}\right|+\left|a_{2}\right|-\left|a_{3}\right|+\cdots+\left|a_{2 n-2}\right|-\left|a_{2 n-1}\right|+\left|a_{2 n}\right|-\left|a_{2 n+1}\right| \\ & =\left(\left|a_{0}\right|-\left|a_{1}\right|\right)+\left(\left|a_{2}\right|-\left|a_{3}\right|\right)+\cdots+\left(\left|a_{2 n-2}\right|-\left|a_{2 n-1}\right|\right)+\left(\left|a_{2 n}\right|-\left|a_{2 n+1}\right|\right) \\ & =S_{2 n-1}+\left(\left|a_{2 n}\right|-\left|a_{2 n+1}\right|\right) \end{aligned}$
и так как $\left|a_{2 n}\right| \geq\left|a_{2 n+1}\right|$ , то $S_{2 n+1} \geq S_{2 n-1}$ , т.е. она не убывает. Итак, последовательность $\left(S_{2 n+1}\right)$ ограничена сверху и не убывает, тогда по теореме Вейерштрасса 1.4 .7 у неё есть предел $\lim _{n \rightarrow \infty} \mathrm{~S}_{2 n+1}=\mathrm{S} \leq\left|a_{0}\right|$ .
Наконец, мы также можем записать $\begin{aligned} S_{2 n+1} & =\left|a_{0}\right|-\left|a_{1}\right|+\left|a_{2}\right|-\left|a_{3}\right|+\left|a_{2 n}\right|-\left|a_{2 n+1}\right| \\ & =S_{2 n}-\left|a_{2 n+1}\right| \end{aligned}$
так как $\lim _{n \rightarrow \infty} \mathrm{~S}_{2 n+1}=\mathrm{S}$ и по условию $\lim _{n \rightarrow \infty}\left|a_{2 n+1}\right|=0$ , то по теореме 1.3.1 $\lim _{n \rightarrow \infty} S_{2 n}=\lim _{n \rightarrow \infty}\left(S_{2 n+1}+\left|a_{2 n+1}\right|\right)=S+0=S$
Итак, мы показали, что $\lim _{n \rightarrow \infty} \mathrm{~S}_{n}=\mathrm{S}$ , что и означает сходимость ряда.

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Problem 718

Доказательство. Воспользовавшись леммой 2.1.10, мы можем считать, что $\left|a_{n}\right| \geq\left|a_{n+1}\right|$ для всех $n$ . Для удобства положим, что первый элемент ряда - это $a_{0}$ , m.e. $n \geq 0$ . Рассмотрим частичную сумму $S_{2 n+1}$ , имеем $\begin{aligned} \mathrm{S}_{2 n+1} & =\left|a_{0}\right|-\left|a_{1}\right|+\left|a_{2}\right|-\left|a_{3}\right|+\left|a_{4}\right|+\cdots+\left|a_{2 n}\right|-\left|a_{2 n+1}\right| \\ & =\left|a_{0}\right|-\left(\left|a_{1}\right|-\left|a_{2}\right|\right)-\left(\left|a_{3}\right|-\left|a_{4}\right|\right)-\cdots-\left(\left|a_{2 n-1}\right|-\left|a_{2 n}\right|\right)-\left|a_{2 n+1}\right| \end{aligned}$
так как $\left|a_{n}\right| \geq\left|a_{n+1}\right|$ , то каждая скобка положительна, это значит, что $S_{2 n+1} \leq\left|a_{0}\right|$ , т.е. последовательность ( $\mathrm{S}_{2 n+1}$ ) ограничена сверху.
С другой стороны, мы можем записать $\begin{aligned} \mathrm{S}_{2 n+1} & =\left|a_{0}\right|-\left|a_{1}\right|+\left|a_{2}\right|-\left|a_{3}\right|+\cdots+\left|a_{2 n-2}\right|-\left|a_{2 n-1}\right|+\left|a_{2 n}\right|-\left|a_{2 n+1}\right| \\ & =\left(\left|a_{0}\right|-\left|a_{1}\right|\right)+\left(\left|a_{2}\right|-\left|a_{3}\right|\right)+\cdots+\left(\left|a_{2 n-2}\right|-\left|a_{2 n-1}\right|\right)+\left(\left|a_{2 n}\right|-\left|a_{2 n+1}\right|\right) \\ & =S_{2 n-1}+\left(\left|a_{2 n}\right|-\left|a_{2 n+1}\right|\right) \end{aligned}$
и так как $\left|a_{2 n}\right| \geq\left|a_{2 n+1}\right|$ , то $S_{2 n+1} \geq S_{2 n-1}$ , т.е. она не убывает. Итак, последовательность $\left(S_{2 n+1}\right)$ ограничена сверху и не убывает, тогда по теореме Вейерштрасса 1.4 .7 у неё есть предел $\lim _{n \rightarrow \infty} \mathrm{~S}_{2 n+1}=\mathrm{S} \leq\left|a_{0}\right|$ .
Наконец, мы также можем записать $\begin{aligned} S_{2 n+1} & =\left|a_{0}\right|-\left|a_{1}\right|+\left|a_{2}\right|-\left|a_{3}\right|+\left|a_{2 n}\right|-\left|a_{2 n+1}\right| \\ & =S_{2 n}-\left|a_{2 n+1}\right| \end{aligned}$
так как $\lim _{n \rightarrow \infty} \mathrm{~S}_{2 n+1}=\mathrm{S}$ и по условию $\lim _{n \rightarrow \infty}\left|a_{2 n+1}\right|=0$ , то по теореме 1.3.1 $\lim _{n \rightarrow \infty} S_{2 n}=\lim _{n \rightarrow \infty}\left(S_{2 n+1}+\left|a_{2 n+1}\right|\right)=S+0=S$
Итак, мы показали, что $\lim _{n \rightarrow \infty} \mathrm{~S}_{n}=\mathrm{S}$ , что и означает сходимость ряда.

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Problem 719

Предложение 2.3.10. Если ряд абсолютно сходится, то при любой перестановке его элементов абсолютная сходимость полученного нового ряда не нарушается и более того, его сумма остаётся прежней.
Доказательство. Пусть ряд $\left(x_{n}\right)$ сходится абсолютно, рассмотрим ряды $\left(x_{n}^{+}\right),\left(x_{n}^{-}\right)$ (конструкция 2.3.8), очевидно, что $x_{n}=x_{n}^{+}-x_{n}^{-}$ для всех $n$ . Так как ряд ( $x_{n}$ ) сходится абсолютно, то ввиду $x_{n}^{+} \leq\left|x_{n}\right|, x_{n}^{-} \leq\left|x_{n}^{-}\right|$ и признака сравнения (теорема 2.2.2) ряды $\left(x_{n}^{+}\right),\left(x_{n}^{-}\right)$ тоже сходятся.
Пусть ряд, полученный после перестановки исходного ряда, имеет вид ( $y_{n}$ ), рассмотрим также ряды $\left(y_{n}^{+}\right),\left(y_{n}^{-}\right)$ (Конструкция 2.3.8), тогда $y_{n}=y_{n}^{+}-y_{n}^{-}$ , и мы получаем $\begin{aligned} S & =S^{+}-S^{-} \\ & =\sum_{n=1}^{\infty} x_{n}^{+}-\sum_{n=1}^{\infty} x_{n}^{-} \\ & =\sum_{n=1}^{\infty} y_{n}^{+}-\sum_{n=1}^{\infty} y_{n}^{-} \\ & =\sum_{n=1}^{\infty}\left(y_{n}^{+}-y_{n}^{-}\right) \\ & =\sum_{n=1}^{\infty} y_{n} . \end{aligned}$ (по предложению 2.3.2) (по теореме 2.2.10) (по предложению 2.1.8)
Что и требовалось доказать.

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Problem 720

DEPARTMENT: SCIENCE LABORATORY TITLE: MTH 101
The $n^{th}$ term of the sequence is given as $3 \cdot 2^{2n}$ . Obtain the first four term of Sequence.

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Problem 721

A finite sequence is shown. $\{-25,-22,-19, \ldots, 32\}$
Which sigma notation can be used to represent the series for the finite sequence? Help: Introduction to Sigma Notation (video). $\sum_{n=1}^{18}(3 n-28)$ $\sum_{n=1}^{20}(3 n-28)$ $\sum_{n=1}^{20}(-3 n-22)$ $\sum_{n=1}^{18}(-3 n-22)$

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Problem 722

```latex \text{Part I-Write the most simplified form of your answer in the space provided (3 points)}
\text{a) Determine whether the following sequence converges or diverges.}
\text{a. } \sum_{n=0}^{\infty} \frac{2^{2}}{(x+1)^{2}} \text{ is } \qquad
\text{b. } \sum_{n=1}^{\infty} \frac{3}{n^{2}-3n+2} \text{ is } \qquad
\qquad \text{convergent, or divergent} \qquad
\text{1. For what values of } x \text{ does the series } \sum_{n=0}^{\infty} n x^{n} \text{ converge?} \qquad
\text{Part II: Work out each of the following clearly and neatly showing all the necessary steps: (4 points each)}
\text{5. Determine if the following series converges or diverges.}
\text{a. } \sum_{i=3}^{\infty}\left(1-\frac{3}{n}\right) m^{2} \text{ by root test.}
\text{b. } \sum^{\infty}=\frac{n}{\sqrt{\pi^{2}-12}} \text{ by comparison test.} ```

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Problem 723

เi. on 5 - Notes C. Q Line Guide Reset Answer
This question has multiple parts. Be sure to answer all the parts of this question.
Each figure is created using green hexagon tiles. PART A
Is the sequence describing the number of green hexagons used in each figure an arithmetic or geometric sequence? Explain.
Figure 1 Figure 2 Figure 3 Enter your response here

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Problem 724

Determine the monthly payment for the installment loan. \begin{tabular}{|l|l|l|l|} \hline Amount & \begin{tabular}{l} Annual \\ Percentage \end{tabular} & \begin{tabular}{l} Number of \\ Payments per \\ Year $(\mathbf{n})$ \end{tabular} & \begin{tabular}{l} Time in \\ Years $(t)$ \end{tabular} \\ \hline Financed (P) & Rate $(\boldsymbol{r})$ & 12 & 3 \\ \hline $\$ 14,000$ & $4 \%$ & & \\ \hline \end{tabular}
Click the icon to view the partial APR table.
The monthly payment is \\square$ (Round to the nearest cent as needed.)

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Problem 725

Determine the monthly payment for the installment loan. \begin{tabular}{|c|c|c|c|} \hline \begin{tabular}{l} Amount \\ Financed (P) \end{tabular} & Annual Percentage Rate (r) & Number of Payments per Year (n) & Time in Years (t) \\ \hline \$14,000 & 4\% & 12 & 3 \\ \hline \end{tabular}
Click the icon to view the partial APR table.
The monthly payment is $\$$ $\square$ (Round to the nearest cent as needed.) Finance Rates

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Problem 726

Question 3 4 Pc
Use the Principle of Mathematical Induction to select the induction step for the proof of the logical statement. $P(n): 2+7+12+\ldots .+(5 n-3)=\frac{n}{2}(5 n-1)$ , for $n \in N$ . Therefore, for some $k \in N$ , the induction step would be given by (A) $\frac{k}{2}(5 k-1)+(5 k+2)=\frac{k+1}{2}(5(k+1)-1)$ (B) $\frac{k}{2}(5 k-1)+k=\frac{k}{2}(5 k-3)$ (C) $(5 \mathrm{k}-3)+(\mathrm{k}+1)=2(3 \mathrm{k}-1)$ (D) $k(5 k-1)+k=5 k^{2}$

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Problem 727

3.12. $f(x)=3-x,-2<x<2, l=2$ . (Oтвет: $3-x=$ $\left.=2+\frac{4}{\pi} \sum_{n=1}^{\infty}\left(\frac{(-1)^{n}}{n} \sin \frac{n \pi x}{2}\right) .\right)$

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Problem 728

Ray Flagg took out a 60 -month fixed installment loan of $\$ 12,000$ to open a new pet store. He paid no money down and began making monthly payments of $\$ 232$ . Ray's business does better than expected and instead of making his 36 th payment, Ray wishes to repay his loan in full. Complete parts a) through c).
Click the icon to view the table of interest rates. a) Determine the APR of the installment loan. $\mathrm{APR}=6.0 \%$ b) How much interest will Ray save by paying off the loan early? (Use the actuarial method)
Interest saved $=\$ 333.44$ (Round to the nearest cent.) c) What is the total amount due to pay off the loan?
The total amount due $=\$$ $\square$ (Round to the nearest cent.) table of interest rates \begin{tabular}{|ccccccccccc|} \hline & \multicolumn{8}{c|}{ Annual Percentage Rate } \\ \cline { 2 - 10 } Number of & $3.0 \%$ & $3.5 \%$ & $4.0 \%$ & $4.5 \%$ & $5.0 \%$ & $5.5 \%$ & $6.0 \%$ & $6.5 \%$ & $7.0 \%$ \\ \hline \multicolumn{7}{c|}{ (Finance charge per $\$ 100$ } & of amount financed) \\ Payments & 3.15 & 3.69 & 4.22 & 4.75 & 5.29 & 5.83 & 6.37 & 6.91 & 7.45 \\ 24 & 3.92 & 4.58 & 5.25 & 5.92 & 6.59 & 7.26 & 7.94 & 8.61 & 9.30 \\ 30 & 4.69 & 5.49 & 6.29 & 7.09 & 7.90 & 8.71 & 9.52 & 10.34 & 11.16 \\ 36 & 6.24 & 7.31 & 8.38 & 9.46 & 10.54 & 11.63 & 12.73 & 13.83 & 14.94 \\ 48 & 7.81 & 9.15 & 10.50 & 11.86 & 13.23 & 14.61 & 16.00 & 17.40 & 18.81 \\ 60 & & & & & & & & \end{tabular}

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Problem 729

اذا كان $f(z)=\frac{e^{z}-e^{-z}}{z^{4}}$ أوجد مايلي:
1. متسلسلة لورانت للاقتران (z) f(z
2. اوجد قيم z بحيث تكون متسلسلة لورانت متقاربة.

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Problem 730

اذا كان $f(z)=\frac{e^{z}-e^{-z}}{z^{4}}$ أوجد مايلي:
1. متسلسلة لورانت للاقتبان (z)
2. اوجد قيم z بحيث تكون متسلسلة لورانت متقاربة.

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Problem 731

Which represents the explicit formula for the arithmetic sequence $a_{n}=15+5(n-1)$ in function form? $f(n)=5 n+10$ $f(n)=n+10$ $f(n)=n+20$ $f(n)=5 n+15$ Previous

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Problem 732

Prove $P_{n}$ : $\sum_{k=1}^{n} \frac{1}{k(k+1)}=\frac{n}{n+1}$ for all positive integers $n$ . What’s the first induction step?

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Problem 733

\text{Show that the series } \sum_{n=1}^{\infty} \frac{1}{n^{\alpha}} \text{ converges.}

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Problem 734

3 8 points
In units of $h R$ , what is the amount of energy associated with the transition from $n=4$ to $n=1$ in the hydrogen emission spectrum? ( h is Planck's constant and $R=3.3 \times 10^{15} \mathrm{~Hz}$ in the Rydberg equation, although you don't need to use these numbers in this problem.) $\frac{1}{16} h R$ $\frac{3}{4} h R$ $\frac{1}{4} h R$ $\frac{15}{16} h R$ 4 8 points

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Problem 735

Considérons la fonction numérique $f$ de la variable réelle $x$ définie sur $\mathbb{R}$ par: $f(x)=\frac{1}{2} \sqrt{x^{2}+3}$
1. a) Montrer que: $f^{\prime}(x)=\frac{x}{2 \sqrt{x^{2}+3}}$ pour tout $x \geq 0$ , puis dresser le tableau de variations de la fonction $f$ sur $\mathbb{R}$ . a) Montrer que : $f([0 ; 1] \subset[0 ; 1[$ .
2. a) Résoudre dans $\mathbb{R}$ l'équation $(E): f(x)=x$ . b) Montrer que : $f(x)>x$ pour tout élément $x$ de l'intervalle $[0 ; 1[$ .
3. On considère la suite $\left(U_{n}\right)$ définie par : $U_{0}=0$ et $U_{n+1}=f\left(U_{n}\right)$ pour tout entier naturel $n$ . 3.1. Montrer par récurrence que : $0 \leq U_{n}<1$ pour tout entier naturel $n$ . 3.2. Etudier le sens de variations de la suite $\left(U_{n}\right)$ . 3.3. Montrer que la suite $\left(U_{n}\right)$ est convergente et déterminer sa limite.

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Problem 736

(20 علامة)

See Solution

Problem 737

( 20 علامة) $\sum_{r=1}^{n} 2 r-1=n^{2}$
السؤال الثالث:
برهن باستخدام الاستقراء الرياضي أن:

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Problem 738

$-30.000+\sum_{t=1}^{6} \frac{8.000}{(1+0,12)^{t}}$

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Problem 739

1. Find $F_{30}$ .
2. Calculate $F_{10} + F_{20} + F_{5}$ .
3. Solve $3 F_{18} - 2 F_{10}$ .
4. Determine $F_{21} - 5 F_{7}$ .
5. Find $\frac{4 F_{5} + F_{11}}{2}$ .
6. Calculate $\frac{3 F_{13}}{6}$ .
7. What is $\mathrm{F}_{75} * 0$ ?
8. Compute $10 \mathrm{~F}_{6} * 3$ .
9. Find $\frac{F_{35}}{2}$ .
10. Calculate $\frac{F_{29}}{3} * 3$ .

See Solution

Problem 740

Find the explicit formula for the arithmetic sequence starting with $a_{1}=26$ and the terms $26,33,40,47,54,61,\ldots$ .

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Problem 741

Question 1 (a) Kapil opened a recurring deposit account in a bank. He deposits ₹ 1500 every month [3] for 2 years at $5 \%$ simple interest per annum. Find the total interest earned by Kapil on maturity. b) If $A=\left[\begin{array}{ll}2 & 1 \\ 1 & 2\end{array}\right], B=\left[\begin{array}{ll}1 & 4 \\ 2 & 3\end{array}\right]$ and $C=\left[\begin{array}{ll}-1 & 2 \\ -2 & 5\end{array}\right]$ , find $A(B-C)$ . [3]
The table below shows the daily expenditure on food of 50 house-holds in a locality. [4] \begin{tabular}{|c|c|c|c|c|c|c|} \hline \begin{tabular}{c} Daily \\ Expenditure \\ (in ₹) \end{tabular} & $0-100$ & $100-200$ & $200-300$ & $300-400$ & $400-500$ & $500-600$ \\ \hline \begin{tabular}{c} Number of \\ House-holds \end{tabular} & 5 & 8 & 15 & 10 & 7 & 5 \\ \hline \end{tabular}
Using graph paper, draw a histogram representing the above distribution and estimate the mode. Take along $x$ -axis $2 \mathrm{~cm}=₹ 100$ and along $y$ -axis $2 \mathrm{~cm}=2$ Households.
This paper consists of 8 printed pages. 11 Turn Ov yright reserved.

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Problem 742

1. Is the series $\sum_{n=1}^{\infty} \frac{n^{1000}}{1.001^{n}}$ convergent or divergent?

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Problem 743

Given two terms in an arithmetic sequence find the recursive formula. 27) $a_{18}=3362$ and $a_{38}=7362$ 28) $a_{18}=44.3$ and $a_{33}=84.8$

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Problem 744

Find the arithmetic sequence for $a_{n}=5n+6$ starting with $n=1$ .

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Problem 745

Find the arithmetic sequence for $n=19$ given the formula $a_{n}=6n+0$ .

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Problem 746

Prove by induction that for positive integers $n$ :
$1 + 2(\frac{1}{2}) + 3(\frac{1}{2})^2 + \ldots + n(\frac{1}{2})^{n-1} = 4 - \frac{n+2}{2^{n-1}}$ .

See Solution

Problem 747

Marissa deposits \$1,000 and adds \$100 monthly for 40 years at a 6.5% annual return. What's her total balance?

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Problem 748

Find fixed points of the sequence defined by $a_{n+1}=\frac{1}{2}\left(a_{n}+\frac{3}{a_{n}}\right)$ .

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Problem 749

Find the $11^{\text {th }}$ term of the sequence defined by $2 n^{2}-3$ .

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Problem 750

Find the first negative term $t$ and an expression for the $n^{\text{th}}$ term of the sequence: 32, 26, 20, 14, 8.

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Problem 751

Write an equation to describe the sequence below. Use $n$ to represent the position of a term in the sequence, where $n=1$ for the first term. $34,-102,306, \ldots$
Write your answer using decimals and integers. $a_{n}=\square(\square)^{n-1}$ Submit

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Problem 752

Find the right number that fits the sequence. $\begin{array}{lllllll}7 & 4 & 10 & 12 & 13 & 36 & ?\end{array}$
49 25 16 15 72 Continue

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Problem 753

Soit le sinte $\mu_{n}$ sefin por $\left\{\begin{array}{l}\mu_{0}=4 \\ \mu_{n+1}=\frac{1}{2}\end{array} \mu_{n}+5\right.$ a) calculer $\mu_{1}, \mu_{2}, \mu_{3}$ . b) justifier que (un) ni erthimethique ni Gcométrique o) expore $v_{n}=\mu_{n}-10$ . a) Montre que $\left(v_{n}\right)(S \cdot G)$ . b) Exprimer $v_{n}$ puis $u_{n}$ enfonction den d) oxpose $S_{n}=\sum_{k=0}^{n-1} v_{k}$ et $S_{n}^{\prime}=\sum_{k=0}^{n-1} \mu_{k}$ a) experime $S_{n}$ pins sn'enfonction dek. b) aluiler $S_{325}$ et $S_{2024}$ .

See Solution

Problem 754

Write an expression to describe the sequence below. Use $n$ to represent the position of a term Questions answered in the sequence, where $n=1$ for the first term. $-65,-64,-63,-62, \ldots$ 13

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Problem 755

Select the correct answer from each drop-down menu.
When she was 20, Liz started saving $\$ 6,000$ a year for retirement. Her goal is to reach $\$ 100,000$ in savings by the time she's 30 . Her account earns $8 \%$ interest per year, compounded annually. Liz $\square$ have saved $\$ 100,000$ by age 30 . She'll $\square$ her goal by about $\square$ Reset Next

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Problem 756

Determine if the sequence $200, 40, 8, \ldots$ is arithmetic, geometric, or neither.

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Problem 757

Determine if the sequence defined by $f(1)=10, f(n)=f(n-1)-1.5$ for $n \geq 2$ is arithmetic, geometric, or neither.

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Problem 758

1. Studiare la monotonia e determinare estremo inferiore e superiore della successione $a_{n}=\frac{\cos (n \pi)+n}{n^{2}+\arctan (n)} \quad \text { per } n \in \mathbb{N} \backslash\{0\}$ specificando se sono minimo e massimo.

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Problem 759

Quiz, December 17, 2024 B1 Is the series $\sum_{n=1}^{\infty} \ln \left(1+\frac{1}{n}\right) \cdot(\sqrt{n+5}-\sqrt{n})$ convergent?

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Problem 760

20. What is the fourth term in the expansion of $(a+b)^{4}$ ? a. $4 a^{3} b$ b. $a^{4}$ c. $6 a^{2} b^{2}$ d. $4 a b^{3}$

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Problem 761

Question Write the following using sigma notation and then evaluate. The sum of the terms $2^{k-1}+4$ for $k=2,3, \ldots 5$ .
Provide your answer below:
$\square$ $\square$ $k=$ $\square$

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Problem 762

4) $L^{-1}\left\{\frac{1}{s-1}+\frac{1}{s+1}\right\}=$ a) $2 \sin (t)$ b) $2 \cos (t)$ c) $2 \sinh (t)$ d) $2 \cosh (t)$

See Solution

Problem 763

The terms of the increasing arithmetic sequence $a_{n}$ are positive. The terms of the increasing geometric sequence $g_{n}$ are positive. The values of the first terms of both sequences are the same, and the values of the fourth terms of both sequences are the same. Which of the following statements describes the values of the second terms of the sequences?
A The second term of the arithmetic sequence must be less than the second term of the geometric sequence.
B The second term of the arithmetic sequence must be greater than the second term of the geometric sequence. (C) The second term of the arithmetic sequence must be the same value as the second term of the geometric sequence. (D) The relationship between the values of the second terms cannot be determined from the given information.

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Problem 764

Describe how to generate the next terms in the pattern: A: $5, 8, 11, 14, 17, \ldots$

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Problem 765

Identify the rule for generating terms in these sequences: a. Pattern A: 5, 8, 11, 14, 17, ... b. Pattern B: $\frac{1}{2}, 1, 2, 4, 8, ...$

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Problem 766

6. Assume a business is deciding whether to invest in a new project that is projected to generate profits of $\$ 90,000$ each year for the next three years. The project start-up costs are $\$ 225,000$ . (A) If the business normally earns 11 percent on its investments, should the business invest? Show/explain. (B) If the business normally earns 5 percent on its investments, should the business invest? Show/explain.

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Problem 767

e) $\sum_{n=1}^{\infty} \frac{7^{n}}{4^{n}-3}$

See Solution

Problem 768

2024-12
1. $\sum_{k=1}^{\infty} \frac{\sqrt{k}+3}{2 \sqrt{k}+1}$ $2 \sum_{k=1}^{\infty} \frac{k^{2}+k}{7 k+1}$
3. $\sum_{k=1}^{\infty} \frac{5 k+4}{4 k+3}$
4. $\sum_{k=1}^{\infty} \frac{k+4}{k^{2}+5}$
5. $\sum_{k=1}^{\infty} \frac{1}{\sqrt{k+7}}$ a. 3 and 5 b. 1 and 3 c. 2 and 4

See Solution

Problem 769

A formula for the sequence $a_{n}: 2,14,98,686, \cdots$ , where $n \geq 0$ is
Select one: $7 \cdot 2^{n}$ $2+7 n$ $2 \cdot 7^{n}$ $7+2^{n}$

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Problem 770

12. If $f(x)=2 \cos x$ . Find $T_{6}(x, \pi)$ .

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Problem 771

The input voltage $\mathrm{X}_{(t)}$ ), and output voltage $\mathrm{Y}_{(t)}$ of an electrical system are sampled simultaneously at regular intervals with the following results. $\begin{array}{r} n=0,1,2, \quad 3, \quad 4,5,6,7,8,9, \ldots \\ X(n T)=15,10,6,2, \quad 1,0,0,0,0,0, \ldots \\ Y(n T)=15,15,7.5,-2.75,-2.5, \cdots, \cdots,-,-\ldots \end{array}$
Calculate the missing values of the output voltage $\mathrm{Y}(\mathrm{nT})$ above. Mint: Assume that $\mathrm{X}(\mathrm{t})=0$ for $\mathrm{t}<0$ , and that only nonzero imputse response samples are $\mathrm{h}(\mathrm{n})$ , for $0 \leq \mathrm{n} \leq 4$ .

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Problem 772

8.1 .00 9.1 .00 10.1 .00 11.1 .00 12.1 .00 13.1 .00 14.1 .00

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Problem 773

Exercice 4 Soit la suite $\left(U_{n}\right)$ définiepar $\left\{\begin{array}{c}U_{0}=\frac{2}{3} \\ U_{n+1}=\frac{1}{2} U_{n}+\frac{n}{2 \sqrt{2}}+\frac{1}{\sqrt{2}}\end{array}\right.$
1. Calculer $U_{1}, U_{2}$ et $U_{3}$
2. On pose: $\forall n \geq 0 \quad V_{n}=U_{n} \sqrt{2}-n$ . a. Calculer $V_{0}, V_{1}$ et $V_{2}$ b. Montrer que $\left(V_{n}\right)$ est une suite géométrique c. Exprimer $\boldsymbol{V}_{\boldsymbol{n}}$ puis $\boldsymbol{U}_{\boldsymbol{n}}$ en fonction de $\boldsymbol{n}$ d. Calculer en fonction de $n: S_{n}=\sum_{k=0}^{k=n} v_{k}$

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Problem 774

Problem A Continue the two sequences of numbers below and find an equation to calculate the $n$ -th value: \begin{tabular}{c|c|c|c|c|c|c|c|c} $\mathbf{n}$ & $\mathbf{1}$ & $\mathbf{2}$ & $\mathbf{3}$ & $\mathbf{4}$ & $\mathbf{5}$ & $\mathbf{6}$ & $\mathbf{7}$ & Equation \\ \hline $a_{n}$ & 2 & 5 & 10 & 17 & 26 & 37 & & \\ \hline $b_{n}$ & 1 & 2 & 8 & 48 & 384 & 3840 & & \end{tabular}

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Problem 775

A new car is worth $\$ 25,000$ . However, it loses $12 \%$ of its value each year due to depreciation. Write an explicit formula describing the value of the car, $a_{n}$ , after $n$ years.

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Problem 776

2. [-/1 Points] DETAILS MY NOTES SCALCET8M 11.8.042. $0 / 3$ Submissions Used
Suppose that the radius of convergence of the power series $\sum c_{n} x^{n}$ is $R$ . What is the radius of convergence of the power series $\sum c_{n} x^{4 n}$ ? $\square$

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Problem 777

12. Here are two sequencen: 1 EXAM
Sequence B a. For sequence $A$ , describe a way to produce each new term from the previous term. take Lle previas term and limes it by 10 b. For sequence B, describe a way to produce each new term from the previous term. c. Write a definition for the $n$ th term of sequence $A$ d. Write a definition for the nth term of sequence B e. If these sequences continue, then which is greater, $A(6)$ or $B(6)$ ? Explain or show how you know.

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Problem 778

A ball dropped from 256 feet bounces up $\frac{1}{4}$ of its fall height. What height on the third bounce?

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Problem 779

$\begin{array}{l}\text { rem Complete the pattern: } \\ \begin{array}{l} 4 \times \square \\ 4 \times=4 \\ 4 \times \\ 4 \times=400 \\ 4 \times \\ 4 \times=4,000 \\ 4 \times=40,000 \\ 4 \times=400,000 \\ 4\end{array}\end{array}$

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Problem 780

um erro inferior a $10^{-2}$
4. Usando o método de ponto fixo, determinar o valor aproximado de $\sqrt[5]{-7}$ com erro inferior a $10^{-2}$ . 3.5

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Problem 781

1.2 Calculate $\sum_{\mathrm{m}-2}^{50}(5-2 \mathrm{~m})$

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Problem 782

Дослідити на рівномірну збіжність: $\sum_{n=1}^{\infty} \frac{1}{1+n^{2} x^{2}} \operatorname{arctg} \frac{x}{n}, x \in \mathbb{R} .$

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Problem 783

2) Theresa adds $\$ 3,000$ to her savings account on the first day of each year. Marcus adds $\$ 3,000$ to his savings account on the last day of each year. They both earn 7.5 percent annual interest. What is the difference in their savings account balances at the end of 34 years? You estimate that you will owe \$48,200 in student loans by the time you graduate. The interest rate is 6.52 percent. If you want to have this debt paid in full within six years, how much must you pay each month?

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Problem 784

Exercice ( 5 pts ) On considère la suite $U$ définie par $\left\{\begin{array}{c}U_{0}=\frac{2}{3} \\ U_{n+1}=\frac{3 U_{n}+2}{2 U_{n}+3} ; \forall n \in I N\end{array}\right.$
1. Calculer $\boldsymbol{U}_{\mathbf{1}} ; \boldsymbol{U}_{\mathbf{2}}$
2. Monter que $\forall n \in I N$ on a : $0 \leq U_{n} \leq 1$
3. On pose $V_{n}=\frac{U_{n}-1^{N}}{U_{n}+1}$ a. Monter que la suite $\left(V_{n}\right)$ est une suite géométrique b. Calculer $\boldsymbol{V}_{\boldsymbol{n}}$ puis $\boldsymbol{U}_{\boldsymbol{n}}$ en fonction de $n$ c. Calculer $S_{n}=\sum_{k=0}^{n} V_{n}$ d. Calculer $\lim V_{n} ; \lim U_{n}$ et $\lim S_{n}$

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Problem 785

Soit $\left\{U_{n}\right\}$ et $\left(V_{n}\right)$ deux suites définies par: $U_{n}=\frac{2^{n}+4 n+3}{2}$ et $V_{n}=\frac{2^{n}-4 n+3}{2}$ On pose $T_{1}=U_{n}+V_{n}$ et $T_{2}=U_{n}-V_{n}$ 1) Montrer que $T_{1}$ est géométrique et que $T_{2}$ est arithmétique ? 2) En déduire $S_{1}$ et $S_{2}$ en fonction de $n$ tels que: $S_{1}=\sum_{K=0}^{n} \boldsymbol{U}_{K}$ et $S_{2}=\sum_{K=0}^{n} V_{K}$

See Solution

Sequences & Series

Problem 701

Problem 702

Use the arithmetic sequence formula to find the 38th number in the sequence: 3,6,9…3,6,9 \ldots3,6,9…

Problem 703

Problem 704

Problem 705

Type the missing number in this sequer 1, 3, 9, 27, 81,

Problem 706

Question 3 (1 point) Imagine you need $425\$ 425$425 to cover unexpected car repairs. You decide to get a payday loan. The payday lender charges a $30\$ 30$30 fee for a two-week loan. If this loan was to be rolled over for an entire year then how much would be owed?Your Answer: □\square□ Answer

Problem 707

Problem 708

An investor deposits \$1200 yearly in an annuity at 7% interest. Find the total value after 1, 2, and 3 years.

Problem 709

Keiko deposits \950annuallyina6950 annually in a 6% annuity. Find values at the end of years 1, 2, and 3: (a) 950annuallyina6V_1,(b), (b) ,(b)V_2,(c), (c) ,(c)V_3$.

Problem 710

Ravi deposits \$625 yearly in an annuity at 8% interest. Find its value at the end of years 1, 2, and 3.

Problem 711

Write a function for the arithmetic sequence of Kaia's savings: $525\$ 525$525, $580\$ 580$580, $635\$ 635$635, $690\$ 690$690.

Problem 712

Problem 713

Problem 714

Problem 715

Problem 716

Problem 717

Problem 718

Problem 719

Problem 720

DEPARTMENT: SCIENCE LABORATORY TITLE: MTH 101The nthn^{th}nth term of the sequence is given as 3⋅22n3 \cdot 2^{2n}3⋅22n. Obtain the first four term of Sequence.

Problem 721

Problem 722

Problem 723

Problem 724

Problem 725

Problem 726

Problem 727

3.12. f(x)=3−x,−2<x<2,l=2f(x)=3-x,-2<x<2, l=2f(x)=3−x,−2<x<2,l=2. (Oтвет: 3−x=3-x=3−x= =2+4π∑n=1∞((−1)nnsin⁡nπx2).)\left.=2+\frac{4}{\pi} \sum_{n=1}^{\infty}\left(\frac{(-1)^{n}}{n} \sin \frac{n \pi x}{2}\right) .\right)=2+π4​n=1∑∞​(n(−1)n​sin2nπx​).)

Problem 728

Problem 729

اذا كان f(z)=ez−e−zz4f(z)=\frac{e^{z}-e^{-z}}{z^{4}}f(z)=z4ez−e−z​ أوجد مايلي:1. متسلسلة لورانت للاقتران (z) f(z2. اوجد قيم z بحيث تكون متسلسلة لورانت متقاربة.

Problem 730

اذا كان f(z)=ez−e−zz4f(z)=\frac{e^{z}-e^{-z}}{z^{4}}f(z)=z4ez−e−z​ أوجد مايلي:1. متسلسلة لورانت للاقتبان (z)2. اوجد قيم z بحيث تكون متسلسلة لورانت متقاربة.

Problem 731

Which represents the explicit formula for the arithmetic sequence an=15+5(n−1)a_{n}=15+5(n-1)an​=15+5(n−1) in function form? f(n)=5n+10f(n)=5 n+10f(n)=5n+10 f(n)=n+10f(n)=n+10f(n)=n+10 f(n)=n+20f(n)=n+20f(n)=n+20 f(n)=5n+15f(n)=5 n+15f(n)=5n+15 Previous

Problem 732

Prove PnP_{n}Pn​: ∑k=1n1k(k+1)=nn+1\sum_{k=1}^{n} \frac{1}{k(k+1)}=\frac{n}{n+1}∑k=1n​k(k+1)1​=n+1n​ for all positive integers nnn. What’s the first induction step?

Problem 733

\text{Show that the series } \sum_{n=1}^{\infty} \frac{1}{n^{\alpha}} \text{ converges.}

Problem 734

Problem 735

Problem 736

(20 علامة)

Problem 737

( 20 علامة) ∑r=1n2r−1=n2\sum_{r=1}^{n} 2 r-1=n^{2}r=1∑n​2r−1=n2السؤال الثالث:برهن باستخدام الاستقراء الرياضي أن:

Problem 738

−30.000+∑t=168.000(1+0,12)t-30.000+\sum_{t=1}^{6} \frac{8.000}{(1+0,12)^{t}}−30.000+∑t=16​(1+0,12)t8.000​

Problem 739

Problem 740

Find the explicit formula for the arithmetic sequence starting with a1=26a_{1}=26a1​=26 and the terms 26,33,40,47,54,61,…26,33,40,47,54,61,\ldots26,33,40,47,54,61,….

Problem 741

Problem 742

1. Is the series ∑n=1∞n10001.001n\sum_{n=1}^{\infty} \frac{n^{1000}}{1.001^{n}}∑n=1∞​1.001nn1000​ convergent or divergent?

Problem 743

Given two terms in an arithmetic sequence find the recursive formula. 27) a18=3362a_{18}=3362a18​=3362 and a38=7362a_{38}=7362a38​=7362 28) a18=44.3a_{18}=44.3a18​=44.3 and a33=84.8a_{33}=84.8a33​=84.8

Problem 744

Find the arithmetic sequence for an=5n+6a_{n}=5n+6an​=5n+6 starting with n=1n=1n=1.

Problem 745

Find the arithmetic sequence for n=19n=19n=19 given the formula an=6n+0a_{n}=6n+0an​=6n+0.

Problem 746

Prove by induction that for positive integers nnn:1+2(12)+3(12)2+…+n(12)n−1=4−n+22n−11 + 2(\frac{1}{2}) + 3(\frac{1}{2})^2 + \ldots + n(\frac{1}{2})^{n-1} = 4 - \frac{n+2}{2^{n-1}}1+2(21​)+3(21​)2+…+n(21​)n−1=4−2n−1n+2​.

Problem 747

Marissa deposits \$1,000 and adds \$100 monthly for 40 years at a 6.5% annual return. What's her total balance?

Problem 748

Find fixed points of the sequence defined by an+1=12(an+3an)a_{n+1}=\frac{1}{2}\left(a_{n}+\frac{3}{a_{n}}\right)an+1​=21​(an​+an​3​).

Problem 749

Find the 11th 11^{\text {th }}11th term of the sequence defined by 2n2−32 n^{2}-32n2−3.

Problem 750

Find the first negative term ttt and an expression for the nthn^{\text{th}}nth term of the sequence: 32, 26, 20, 14, 8.

Problem 751

Problem 752

Use the arithmetic sequence formula to find the 38th number in the sequence: $3,6,9 \ldots$

Question 3 (1 point) Imagine you need $\$ 425$ to cover unexpected car repairs. You decide to get a payday loan. The payday lender charges a $\$ 30$ fee for a two-week loan. If this loan was to be rolled over for an entire year then how much would be owed?
Your Answer: $\square$ Answer

Keiko deposits \ $950 annually in a 6% annuity. Find values at the end of years 1, 2, and 3: (a)$ V_1 $, (b)$ V_2 $, (c)$ V_3$.

Write a function for the arithmetic sequence of Kaia's savings: $\$ 525$ , $\$ 580$ , $\$ 635$ , $\$ 690$ .

DEPARTMENT: SCIENCE LABORATORY TITLE: MTH 101
The $n^{th}$ term of the sequence is given as $3 \cdot 2^{2n}$ . Obtain the first four term of Sequence.

3.12. $f(x)=3-x,-2<x<2, l=2$ . (Oтвет: $3-x=$ $\left.=2+\frac{4}{\pi} \sum_{n=1}^{\infty}\left(\frac{(-1)^{n}}{n} \sin \frac{n \pi x}{2}\right) .\right)$

اذا كان $f(z)=\frac{e^{z}-e^{-z}}{z^{4}}$ أوجد مايلي:
1. متسلسلة لورانت للاقتران (z) f(z
2. اوجد قيم z بحيث تكون متسلسلة لورانت متقاربة.

اذا كان $f(z)=\frac{e^{z}-e^{-z}}{z^{4}}$ أوجد مايلي:
1. متسلسلة لورانت للاقتبان (z)
2. اوجد قيم z بحيث تكون متسلسلة لورانت متقاربة.

Which represents the explicit formula for the arithmetic sequence $a_{n}=15+5(n-1)$ in function form? $f(n)=5 n+10$ $f(n)=n+10$ $f(n)=n+20$ $f(n)=5 n+15$ Previous

Prove $P_{n}$ : $\sum_{k=1}^{n} \frac{1}{k(k+1)}=\frac{n}{n+1}$ for all positive integers $n$ . What’s the first induction step?

( 20 علامة) $\sum_{r=1}^{n} 2 r-1=n^{2}$
السؤال الثالث:
برهن باستخدام الاستقراء الرياضي أن:

$-30.000+\sum_{t=1}^{6} \frac{8.000}{(1+0,12)^{t}}$

Find the explicit formula for the arithmetic sequence starting with $a_{1}=26$ and the terms $26,33,40,47,54,61,\ldots$ .

1. Is the series $\sum_{n=1}^{\infty} \frac{n^{1000}}{1.001^{n}}$ convergent or divergent?

Given two terms in an arithmetic sequence find the recursive formula. 27) $a_{18}=3362$ and $a_{38}=7362$ 28) $a_{18}=44.3$ and $a_{33}=84.8$

Find the arithmetic sequence for $a_{n}=5n+6$ starting with $n=1$ .

Find the arithmetic sequence for $n=19$ given the formula $a_{n}=6n+0$ .

Prove by induction that for positive integers $n$ :
$1 + 2(\frac{1}{2}) + 3(\frac{1}{2})^2 + \ldots + n(\frac{1}{2})^{n-1} = 4 - \frac{n+2}{2^{n-1}}$ .

Find fixed points of the sequence defined by $a_{n+1}=\frac{1}{2}\left(a_{n}+\frac{3}{a_{n}}\right)$ .

Find the $11^{\text {th }}$ term of the sequence defined by $2 n^{2}-3$ .

Find the first negative term $t$ and an expression for the $n^{\text{th}}$ term of the sequence: 32, 26, 20, 14, 8.

Find the right number that fits the sequence. $\begin{array}{lllllll}7 & 4 & 10 & 12 & 13 & 36 & ?\end{array}$
49 25 16 15 72 Continue

Write an expression to describe the sequence below. Use $n$ to represent the position of a term Questions answered in the sequence, where $n=1$ for the first term. $-65,-64,-63,-62, \ldots$ 13

Determine if the sequence $200, 40, 8, \ldots$ is arithmetic, geometric, or neither.

Determine if the sequence defined by $f(1)=10, f(n)=f(n-1)-1.5$ for $n \geq 2$ is arithmetic, geometric, or neither.

Quiz, December 17, 2024 B1 Is the series $\sum_{n=1}^{\infty} \ln \left(1+\frac{1}{n}\right) \cdot(\sqrt{n+5}-\sqrt{n})$ convergent?

20. What is the fourth term in the expansion of $(a+b)^{4}$ ? a. $4 a^{3} b$ b. $a^{4}$ c. $6 a^{2} b^{2}$ d. $4 a b^{3}$

Question Write the following using sigma notation and then evaluate. The sum of the terms $2^{k-1}+4$ for $k=2,3, \ldots 5$ .
Provide your answer below:
$\square$ $\square$ $k=$ $\square$

4) $L^{-1}\left\{\frac{1}{s-1}+\frac{1}{s+1}\right\}=$ a) $2 \sin (t)$ b) $2 \cos (t)$ c) $2 \sinh (t)$ d) $2 \cosh (t)$

Describe how to generate the next terms in the pattern: A: $5, 8, 11, 14, 17, \ldots$

Identify the rule for generating terms in these sequences: a. Pattern A: 5, 8, 11, 14, 17, ... b. Pattern B: $\frac{1}{2}, 1, 2, 4, 8, ...$

e) $\sum_{n=1}^{\infty} \frac{7^{n}}{4^{n}-3}$

A formula for the sequence $a_{n}: 2,14,98,686, \cdots$ , where $n \geq 0$ is
Select one: $7 \cdot 2^{n}$ $2+7 n$ $2 \cdot 7^{n}$ $7+2^{n}$

12. If $f(x)=2 \cos x$ . Find $T_{6}(x, \pi)$ .