Browse Sequences & Series problems Studdy Solved!

Problem 201

12. Which explicit rule matches the sequence described in the table? \begin{tabular}{|l|c|c|c|c|} \hline $\boldsymbol{n}$ & 1 & 3 & 5 & 7 \\ \hline $\boldsymbol{f}(\boldsymbol{n})$ & -1 & -5 & -9 & -13 \\ \hline \end{tabular}

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Problem 202

A biologist predicts the number of bacteria in a sample after $n$ hours using the sequence below. Use the drop-down menus to describe the sequence. $50,200,800,3200, \ldots$
Click the arrows to choose an answer from each menu.
The sequence has an initial term of Choose... $\square$ and a common Choose... $\square$

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Problem 203

Enter the first five terms of the following recursively defined sequence. $\begin{array}{l} a_{1}=1 ; a_{n}=a_{n-1}+n, n \geq 2 \\ a_{1}=\square \end{array}$ $\square$ $a_{2}=$ $\square$ $a_{3}=$ $\square$ $a_{4}=$ $\square$ $a_{5}=\square$

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Problem 204

Find an explicit formula for the $n^{\text {th }}$ term of this sequence. $\begin{array}{l} 9,-1, \frac{1}{9},-\frac{1}{81}, . . \\ a_{n}= \end{array}$

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Problem 205

Use the given information to write the first 5 terms of the arithmetic sequence. $a_{14}=-114, a_{24}=-204$ $\begin{array}{l} a_{1}=\square \\ a_{2}=\square \\ a_{3}=\square \\ a_{4}=\square \\ a_{5}=\square \end{array}$

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Problem 206

En una secuencia, cada número se forma sumando los dos de arriba. ¿Qué valor tiene $x$ si 26 = 10 + 16?

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Problem 207

Arithmetic Progression Questions:
1. Is this an AP? $2, -2, -6, -10, \ldots$
2. Next two terms of $3, \frac{7}{2}, 4, \frac{9}{2}, \ldots$ ?
3. Common difference of $2+\sqrt{5}, 2+3\sqrt{5}, 2+5\sqrt{5}, \ldots$ ?
4. Sum of first 500 natural numbers?

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Problem 208

How much will Cardo save on the 9th day if he starts with \$75 and doubles his savings daily?

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Problem 209

Cari sebutan ke- $n$ , $T_{n}$ , dari pola: $5=3+2(1)$ , $9=3+3(2)$ , $15=3+4(3)$ .

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Problem 210

Find the common ratio $r$ of the geometric sequence 7, 189, $y$ , 1701. Then, determine the smallest integer $n$ for which the $n$ -th term exceeds 10,000.

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Problem 211

Solve the equation: $n \cdot n! - (n+1)! = -n!$ for integer values of $n$ .

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Problem 212

Solve the equation $(n+1)^{2} \cdot n !+(n+1) !=(n+2) !$ for n.

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Problem 213

Find the 37th term of an arithmetic sequence where $t_{1}=-3$ and $t_{2}=11$ .

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Problem 214

Find the sum of the first 31 terms of an arithmetic series where $t_{1}=-9$ and $t_{2}=9$ .

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Problem 215

Rank the sums $\mathrm{S}_{30}$ of these series from least to greatest: $4+15+26+\ldots$ , $77+79+81+\ldots$ , $12+17+22+\ldots$ , $-178-154-130-\ldots$ .

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Problem 216

A 0.1 kg ball loses 25% of its energy when bouncing. Starting at 1 m, how many bounces until it bounces ≤ 0.25 m?

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Problem 217

Lindiwe has R80 000 invested at $13.75\%$ p.a. and withdraws R25 000 yearly. How many years will this last?

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Problem 218

Find the number of dots in the 48th figure: $4(48)+2$ . Is there a figure with 200 dots? Explain. Total dots from 10th to 30th figure: $S=2m(m+1)$ .

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Problem 219

Parents plan to deposit \$400 on their son's 10th birthday, increasing by \$50 yearly.
a. Find the deposit on his 20th birthday. b. Calculate total deposits by his 20th birthday.

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Problem 220

You sold a bike for \$10,000 and invest it at 4% compounded monthly. How much can you withdraw monthly for 2.5 years?

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Problem 221

Two companies produce phones. Company A starts at 600 and increases by 300 weekly. Company B starts at 500 and increases by $15\%$ weekly. Find: a) Phones produced in week 15 b) Total production in 15 weeks

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Problem 222

A geometric sequence has first term 32 and ratio $\frac{1}{2}$ . a. Find the first 5 terms. b. Write an explicit formula for the $n$ th term of the sequence.

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Problem 223

11/13/24, 1:23 PM Arithmetic Sequences-Joshua Dupree
9. An auditorium has 52 seats in the first row, 57 seats in the second row, 62 seats in the third row, and so on. Find the general term of this arithmetic sequence and the number of seats in the twentieth row.
The general term $a_{n}$ of this arithmetic sequence is $\square$ (Simplify your answer.)
The number of seats in the twentieth row is $\square$ .
10. Jose takes a job that offers a monthly starting salary of $\$ 2300$ and guarantees him a monthly raise of $\$ 150$ during his first year of training Find the general term of this arithmetic sequence and his monthly salary at the end of his training.
The general term of this arithmetic sequence is $\square$ Jose's monthly salary at the end of his training is $\$$ $\square$

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Problem 224

2ตรบกเณ์ ต (1) (ก) ศณูตาสญบูกก $z=1+i+i^{2}+i^{3}+\ldots+i^{2017}$ (8) กณกต $B=1+i+i^{2}+i^{3}+i^{4}+\ldots+i^{100}$

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Problem 225

Question 11. Write the following sum as a simplified algebraic expression in $p$ and $q$ $\sum_{j=3}^{7}(-1)^{j}(3 p j+5 q)$
Sum $=$ $\square$ $?$

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Problem 226

Question 13. Given the following arithmetic sequence $3,5,7, \ldots$ , find the value of the 14 th term and the sum of the first 28 terms. Enter your answers as integers in the appropriate boxes: (a) The 14th term is $\square$ (b) The sum of the first 28 terms is $\square$

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Problem 227

Question 19.
Consider the following geometic series, $S$ , where: $S=9+9(-0.8)+9(-0.8)^{2}+9(-0.8)^{3} \cdots$
Write down the first term, $a$ and the common ratio, $r$ in the boxes below. Enter $a$ : $\square$ Enter $r$ : $\square$
Hence calculate the sum, $S$ and enter your result in the box below. Enter $S$ (to three decimal places) here: $\square$

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Problem 228

and the inter $\sum_{n=0}^{\infty} \frac{(9 x)^{n}}{n!}$

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Problem 229

Given two terms in an arithmetic sequence find the specified term. 1) $a_{17}=-61$ and $a_{10}=-139$
Find $a_{23}$
Given two terms in a geom etric sequence find the specified term. 2) $a_{1}=-2$ and $a_{6}=64$
Find $a_{11}$
Determine the number of terms $\boldsymbol{n}$ in the geometric series. 3) $4+12+36+108 \ldots . S_{n}=39364$
Rewrite the series using sigma notation. 4) $\frac{1}{16}+\frac{1}{32}+\frac{1}{64}+\frac{1}{128}$

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Problem 230

Question 3 [20 marks] A wind farm operates turbines in a sequence where their power output forms an arithmetic pattern based on operational adjustments. The power output of each turbine at specific intervals follows this arithmetic sequence. The sum of the fourth and eighth turbines' power ( $u_{3}+u_{7}$ )output is 10 MW . The product of the first and sixth turbines' power output ( $u_{0} \times u_{5}$ ) is 20 MW . Find the difference, $d$ , between the power outputs of consecutive turbines.

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Problem 231

Determine the number of terms $n$ in the geometr 3) $4+12+36+108 \ldots, S_{n}=39364$

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Problem 232

Find each sum for the following series to the nearest tenth. Type DNE if the sum does not exist. $\begin{array}{l} 256+64+16+\cdots \\ S_{4}=\square \\ S_{\infty}=\square \end{array}$

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Problem 233

Find the exact sum. Type your answer as a decimal, if necessary. $\sum_{n=1}^{\infty} 7.4 \cdot 0.2^{n-1}$

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Problem 234

Find the sum of each of the geometric series given below. For the value of the sum, enter an expression that gives the exact value, rather than en an approximation. A. $-9+3-\frac{3}{3}+\frac{3}{9}-\frac{3}{27}+\frac{3}{81}-\cdots=$ $\square$ B. $\sum_{n=5}^{12}\left(\frac{1}{2}\right)^{n}=$ $\square$

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Problem 235

5th
Work out a the $n$ th term for the number of shaded squares. $n$ th term $=3 n-2$ $\mathbf{b}$ the $\boldsymbol{n}$ th term for the number of white squares. $n$ th term $=n-1$ $c$ the $n$ th term for the total number of squares. $n$ th term $=4 n-3$

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Problem 236

Ministère de l'éducation Nationale Académique Provinciale de la Ngounié Lycée Paul Marie YEMBIT de Ndendé B.P. 03 Ndendé Année scol
Département de Mathématiques
DEVOIR DE MATHEMATIQUES N ${ }^{\circ} 1$ Consignes: Présentation: 01 Date: 18 I Deure: 03 - Aucun échange entre élèves ne sera toléré ; - Les calculatrices sont acceptées; - La présentation sera prise en compte.
Exercice 1:
1. Soit a et $b$ deux réels vérifiant: $0 \leq a<b$
Démontrer les relations: a) $a<\sqrt{a b}<b$ b) $a<\frac{2 a b}{a+b}<\frac{a+b}{2}$
2. Soit $\left(a_{n}\right)$ et ( $b_{n}$ ) deux suites définies pour $n \geq 1$ par: $b_{1}=2 \sqrt{3} \text { et } b_{n+1}=\frac{2 a_{n} b_{n}}{a_{n} b_{n}} \text { puis } a_{1}=3 \text { et } a_{n+1}={\sqrt{a_{n}} b_{n+1}}^{2}$
En utilisant le 1.a, démontrer par récurrence que pour tout $n \geq 1,: 0 \leq a_{n}<b_{n}$
3. En déduire le sens de variation des suites $\left(a_{n}\right)$ et $\left(b_{n}\right)$
4. Montrer la convergence des suites $\left(a_{n}\right)$ et $\left(b_{n}\right)$ .
5. Démontrer que, pour $n \geq 1$ (on pourra utiliser 1.) $b_{n+1}-a_{n+1} \leqslant 1 / 2\left(b_{n}-a_{n}\right)$
6. En déduire que pour $n \geq 1, b_{n}-a_{n} \leq 1 / 2$
7. En déduire que, pour $n \geq 1$ les suites $\left(a_{n}\right)$ et $\left(b_{n}\right)$ convergent vers une même limite. EXERCICE 2

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Problem 237

范 $\qquad$ $\qquad$ - - * $\qquad$ - $\qquad$ $\qquad$ . $4,19,8,24,14,29,19,34,24,39,29,44, \ldots$ Rule: $\qquad$ $\qquad$

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Problem 238

New tal codes f 13 High Econ 10 https://www.webassign.net/web/Student/Assignment-Responses, $\begin{array}{c} f(x)=\frac{1}{(9+x)^{2}} \\ f(x)=\sum_{n=0}^{\infty}\left(\frac{(-1)^{n+1 n x^{n-1}}}{9^{n+1}}\right) \end{array}$
What is the radius of convergence, $R$ ? $R=9$ (b) Use part (a) to find a power series for $\begin{array}{c} f(x)=\frac{1}{(9+x)^{3}} \\ f(x)=\sum_{n=0}^{\infty}(\square) \end{array}$
What is the radius of convergence, $R$ ? $R=$ $\square$ (c) Use part (b) to find a power series for $\begin{array}{c} f(x)=\frac{x^{2}}{(9+x)^{3}} \\ f(x)=\sum_{n=2}^{\infty}(\square) \end{array}$
What is the radius of convergence, $R$ ? $R=$ $\square$ Search

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Problem 239

Find a power series representation for the function. (Give $y$ $\begin{array}{c} f(x)=\ln (5-x) \\ f(x)=\ln (5)-\sum_{n=1}^{\infty}(\square) \end{array}$
Determine the radius of convergence, $R$ . $R=$ $\square$ Need Help? Read It Watch It Master It Submit Answer

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Problem 240

7. [-/2 Points] DETAILS MY NOTES SCALCET9 11.9.02
A graphing calculator is recommended. Find a power series representation for $f$ . (Give your power series repre $\begin{array}{r} f(x)=\ln \left(\frac{1+x}{1-x}\right) \\ f(x)=\sum_{n=0}^{\infty}(\square) \end{array}$
Graph $f$ and several partial sums $s_{n}(x)$ on the same screen.

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Problem 241

he partern. $81-7=74$ 2 $\qquad$ $2+7-7=67$ $72-63=9 \quad 71+9=81$ $90+9 * 1$ he strings II white s for the at is a rule
6. In artist is arranging tiles in rows in decorate a wall. (Aach new row has 2 fewer tiles than the row below it. If the first row has 23 tiles. how many tiles will be in the seventh row?

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Problem 242

7. Write Math Give an example of a rule for a pattern. List a set of numbers that fit the pattern.

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Problem 243

Find the 10th and 15th terms of the sequence $3, 7, 11, 15, 19, \ldots$

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Problem 244

Find the 7th and $n$ th terms of the sequence $5, 3, 1, -1, \ldots$

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Problem 245

Find the $r$ th term of the sequence: 72, 70, 68, 66, ...

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Problem 246

Find the fifteenth term of an A.P. where the 3rd term is 18 and the 7th term is 30.

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Problem 247

Find the $n$ th term of an A.P. where the 5th term is 38 and the 10th term is 23.

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Problem 248

Find the term number of the sequence $5, 14, 23, 32, \ldots$ that equals $239$ .

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Problem 249

Dos corredores en una pista circular se encuentran cada cuánto tiempo si uno tarda $3 \mathrm{~min}$ y el otro $5 \mathrm{~min}$ ?

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Problem 250

Check if these sequences are bounded:
1. $a_{n}=2+3 \cdot n$
2. $a_{n}=n-1$
3. $a_{n}=1-2 \cdot n$
4. $a_{n}=1$
5. $a_{n}=-5 \cdot n$
6. $a_{n}=0$

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Problem 251

Gegeben sind zwei Glieder einer geometrischen Folge. Finde $b_{0}$ , den Quotienten $q$ und die Formel für $b_{n}$ .
a) $b_{1}=98 ; b_{2}=686$ b) $b_{2}=216 ; b_{4}=7776$ c) $b_{3}=1875 ; b_{5}=46875$

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Problem 252

Untersuche, ob die Zahlenfolgen geometrisch sind. Wenn ja, gib die Formel an. Wenn nicht, wie kann man sie ändern?
a) $b_0 = 1$ , $b_1 = -2$ , $b_2 = 4$ b) $b_2 = 100$ , $b_4 = 4$ , $b_6 = 0.25$ c) $b_0 = 8$ , $b_2 = 18$ , $b_5 = 60.75$ d) $b_1 = -1$ , $b_2 = 1$ , $b_5 = 1$

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Problem 253

אורך הצלע הקצרה במלבן הרביעי בסדרה הוא מה? הסדרה מתחילה עם 1, 3, ו-6, והאורך מוכפל ב-2 בכל מלבן.

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Problem 254

Gib die rekursive Darstellung der Folge $x_{n}$ für $x_{n}=2^{n+1}$ , $x_{n}=2n+1$ , und $x_{n}=(-1)^{n}$ an!

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Problem 255

Which function describes the sequence: -5, -7, -9, -11, -13, ...? A) $f(x)=-3x+2$ B) $f(x)=-2x+3$ C) $f(x)=-3x-2$ D) $f(x)=-2x-3$

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Problem 256

Find the recursive formula for the sequence defined by $t_{n}=-5+(n-1) 78$ , with $n \in N$ and $n \geq 1$ .

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Problem 257

Find the next 3 terms of the sequence with $a_{1} = -12$ and $a_{n} = a_{n-1} + 9$ . Choices: A) -3,-15,-27 B) -3,6,15 C) -3,-6,-3 D) -3,-6,3 E) -3,3,9

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Problem 258

Find the next 3 terms in the sequence where $a_{n}=a_{n-1}+9$ and $a_1=-12$ . Choices: -3,-15,-27; B -3,6,15; -3,-6,-3; D -3,-6,3; -3,3,9.

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Problem 259

Identify the function for the sequence $-4, 0, 4, 8, 12, 16, \ldots$ from the options given.

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Problem 260

Identify the version of the sequence with an error among: 1) $f(x)=6 x+5$ , 2) $a_{1}=11$ , $a_{n+1}=a_{n}+6$ , 3) $a_{n}=11+6(n-1)$ . Fix it.

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Problem 261

Which function represents the sequence $-4,0,4,8,12,16,\ldots$ ? A) $f(x)=4x+8$ B) $f(x)=4x-8$ C) $f(x)=-4x+8$ D) $f(x)=-4x-8$

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Problem 262

Which function represents the sequence $3, 7, 11, 15, 19, 23, \ldots$ ? A) $f(x)=4x-4$ , B) $f(x)=4x-3$ , C) $f(x)=4x-2$ , D) $f(x)=4x-1$ .

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Problem 263

Which TWO sequences are equivalent? A: $f(x)=7x-15$ B: $f(x)=7x-22$ C: $f(x)=7x-29$ D: $a_{1}=-22$ , $a_{n+1}=a_{n}+7$ E: $a_{1}=-29$ , $a_{n+1}=a_{n}+7$

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Problem 264

Given positive numbers $a, b$ , determine bounds and find supremum and infimum for these sets:
1. $\{a+b n ; n \in \mathbb{N}\}$
2. $\{a+(-1)^{n} b ; n \in \mathbb{N}\}$
3. $\{a+b / n ; n \in \mathbb{N}^{}\}$
4. $\{(-1)^{n} a+b / n ; n \in \mathbb{N}^{}\}$
5. $\{a+(-1)^{n} b / n ; n \in \mathbb{N}^{*}\}$ .

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Problem 265

Find the number of seats in each row if rows have odd seats, increase by 2, and front row has 1/4 of back rows' total.

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Problem 266

Calculate the sum: $\sum_{n=2}^{6} \frac{n-1}{n(n+1)}$ .

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Problem 267

Find the first three terms of a geometric sequence where the $5^{\text{th}}$ term is 1875 and the $7^{\text{th}}$ term is 46875.

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Problem 268

Determine the common difference in the arithmetic sequence $11, 20, 29, \ldots$

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Problem 269

Überprüfe die Konvergenz der geometrischen Reihen und gib die Summen an: a) $1+2+2^{2}+\ldots$ , b) $1+\frac{1}{2}+\left(\frac{1}{2}\right)^{2}+\ldots$ , c) $1+\left(-\frac{1}{2}\right)+\left(-\frac{1}{2}\right)^{2}+\ldots$ , d) $1+(-1)+(-1)^{2}+\ldots$ , e) $3+3 \cdot \frac{1}{4}+3 \cdot\left(\frac{1}{4}\right)^{2}+\ldots$ , f) $5-5 \cdot 0,1+5 \cdot 0,1^{2}-5 \cdot 0,1^{3}+\ldots$ .

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Problem 270

Berechne das Guthaben nach 10 Jahren, wenn jährlich 5 mal $1000 €$ und 3 mal $800 €$ bei 0,7\% Zinsen eingezahlt werden.

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Problem 271

A Tik Tokker gains 20 more subscribers daily, starting with 100 on day 1. How many total subscribers after 5 days?

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Problem 272

1) Find the expression for the sequence: -66, -132, -198, -264 with $n=1$ for the first term. $a_n=$
2) Find the expression for the sequence: 7, 14, 21, 28 with $n=1$ for the first term. $a_n=$

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Problem 273

3) Find the expression for the sequence -65, -64, -63, ... and calculate the $78^{\text{th}}$ term, $a_{78}$ .
4) Write the expression for the sequence 55, 110, ... and determine the $92^{\text{nd}}$ term, $a_{92}$ .

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Problem 274

What is the partial sum of the arithmetic sequence where $a=17$ , $d=-3$ , and $n=47$ ?

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Problem 275

Find the partial sum of the geometric sequence given $a=3, r=2$ , $\mathrm{n}=8$ .

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Problem 276

$\ln \left(1-5 x^{7}\right)=\sum_{n=1}^{\infty}$ $\square$ For what radius $R$ is the expansion valid? This means that if $|x-0|<R$ , then the Taylor Series converges.
Note: If you use decimals, make sure your answer is correct to at least three decimal places. $R=$ $\square$ If you don't get this in 3 tries, you can get a hint.

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Problem 277

(1 point)
Use the third-order Taylor polynonial for $e^{x} \sin (3 x)$ at $x=0$ to approximate $e^{\frac{1}{t}} \sin (3 / 8)$ by a rational number. $\mathrm{e}^{\frac{1}{1}} \sin (3 / 8) \approx 27 / 256$
Preview My Answers Submit Answers You have attempted this problem 10 times. Your overall recorded score is $0 \%$ . You have unlimited attempts remaining.

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Problem 278

16. $\binom{23}{0}-\binom{23}{1}+\binom{23}{2}-\ldots+\binom{23}{22}-\binom{23}{23}$ ifadesinin değeri kaçtır? A) 0 B) 1 C) 23 D) $22!$ E) $2^{22}$

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Problem 279

Find $\sqrt[4]{\left|a_{k}\right|}$ for the series. $\sum_{k=0}^{\infty}\left(\frac{k}{k+13}\right)^{k}$ (Express numbers in exact form. Use symbolic notation and fractions where needed.)

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Problem 280

Identify the formula to find the sum of the first $n$ terms in a geometric sequence. (A) $s_{n}=\frac{a_{1} \cdot r^{n-1}}{1-r}$ (B) $s_{n}=\frac{a_{1}\left(1-r^{n-1}\right)}{r-1}$ (C) $s_{n}=\frac{a_{1}\left(r^{n}-1\right)}{1-r}$ (D) $s_{n}=\frac{a_{1}\left(1-r^{n}\right)}{1-r}$

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Problem 281

Trovare lo sviluppo di Maclaurin di ordine 3 per la funzione $f(x) = \frac{1}{-\sqrt{8} \sin x - 2 \cos x}$ .

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Problem 282

22 In den Gletschern der Erde sind ungefähr $75 \%$ des lebensnotwendigen Süßwassers gespeichert. Gletscher werden vom Schnee im hochgelegenen Teil genährt und verlieren Wasser im unteren Teil durch Abschmelzen. Dadurch werden Flüsse gespeist. Durch die Klimaerwärmung ist das Ernähren und Abschmelzen nicht im Gleichgewicht: Gletscher verlieren an Größe. Im Jahr 1850 hatten die Gletscher in der Schweiz zusammen eine Fläche von 1800 km² $^{2}$ . Bis zum Jahr 2000 waren davon $750 \mathrm{~km}^{2}$ abgeschmolzen. a) Berechnen Sie, um wie viel Prozent die Gletscherfläche im Mittel pro Jahr abgenommen hat, wenn das Abschmelzen exponentiell erfolgt ist. b) Berechnen Sie, wie groß die Fläche der Schweizer Gletscher im Jahr 2050 sein wird, wenn sich das Abschmelzen in gleicher Weise fortsetzt. c) Durch die in den letzten Jahren deutlich spürbare globale Erwärmung hat sich das Gletschersterben beschleunigt. Man geht davon aus, dass die Gletscherfläche ab dem Jahr 2000 im Mittel pro Jahr um 2,7\% exponentiell abgenommen hat. Berechnen Sie, wie groß die Fläche der Schweizer Gletscher im Jahr 2050 nur noch sein wird, wenn es nicht gelingt, die globale Erwärmung zu verlangsamen oder zu stoppen.
Der Aletsch-Gletscher in den 1930erJahren
Der Aletsch-Gletscher 2015 (im Vergleich mit deutlich verringerter Dicke)

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Problem 283

Y Score: 0/5 Penalty: none 2024/2025 School Year Home Announcements Assignments Modules Clever Log In DeltaMath Desmos calculator Question What is a formula for the nth term of the given sequence? 15, 21, 27... Answer Oan = 15-6(n − 1) © ɑn an = 21-6n an = 21+ 6(n - 1) an = 15+ 6(n - 1) ɑn Copyright ©2024 DeltaMath.com All Rights Reserved. Privacy Policy | Terms of Service

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Problem 284

Question
Find the 63 rd term of the arithmetic sequence $2,9,16, \ldots$ Answer - Attempt 1 out of 2

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Problem 285

3. Ряд $\sum_{n=1}^{\infty} a_{n} \epsilon$ збіжним, а ряд $\sum_{n=1}^{\infty} b_{n}$ є розбіжними. Що можна сказати про збіжність ряду $\sum_{n=1}^{\infty}\left(a_{n}+b_{n}\right) ?$

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Problem 286

4. Відомо, що $a_{n} \geq 0$ для всіх $n \in \mathbb{N}$ , і що ряд $\sum_{n=1}^{\infty} a_{n}^{2}$ збігається. Чи обов'язково ряд $\sum_{n=1}^{\infty} a_{n}$ також збігається?

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Problem 287

Match each of the following with the correct statement. C stands for Convergent, D stands for Divergent.
C 1. $\sum_{n=3}^{\infty} \frac{2}{n^{3}-9}$
2. $\sum_{n=1}^{\infty} \frac{1}{5+\sqrt[4]{n^{7}}}$
3. $\sum_{n=1}^{\infty} \frac{2}{n(n+4)}$
4. $\sum_{n=1}^{\infty} \frac{\ln (n)}{3 n}$
5. $\sum_{n=1}^{\infty} \frac{5+2^{n}}{5+1^{n}}$

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Problem 288

Arithmetic and Geometric
The first term of an arithmetic sequence is 5 , and the common difference of the sequence is 2 . What is eighth term of the sequence?

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Problem 289

ether the following series converges or diverges. $\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{\sqrt{n}}$ nvergence and $D$ for divergence:

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Problem 290

4. Find the Taylor series for $f$ centered at 4 if $f^{(n)}(4)=\frac{(-1)^{n} n!}{3^{n}(n+1)}$
What is the radius of convergence of the Taylor series?

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Problem 291

In practice, we take $-M$ to be the maximum of $\left|f^{(n+1)}(x)\right|$ on the interval $I$ .
Question: (a) Approximate $\ln (1.5)$ using the 3-rd degree Taylor polynomial of $f(x)=\ln (x)$ centred at $x=1$ (b) Use Taylor's Remainder Theorem to approximate the error between $\ln (1.5)$ and the 3 -rd degree Taylor polynomial approximation on the interval $I=\left[\frac{1}{2}, \frac{3}{2}\right]$ . (c) Let $T_{n}(x)$ be the $n$ -th degree Taylor polynomial for $f(x)=\ln (x)$ centred at $x=1$ . Using Taylor's Remainder Theorem, find the smallest integer $n$ so that $\left|f(x)-T_{n}(x)\right| \leq 0.0001$ for all $x \in\left[\frac{1}{2}, \frac{3}{2}\right]$ .

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Problem 292

Question: (a) Approximate $\ln (1.5)$ using the 3-rd degree Taylor polynomial of $f(x)=\ln (x)$ centred at $x=1$ . (b) Use Taylor's Remainder Theorem to approximate the error between $\ln (1.5)$ and the 3 -rd degree Taylor polynomial approximation on the interval $I=\left[\frac{1}{2}, \frac{3}{2}\right]$ . (c) Let $T_{n}(x)$ be the $n$ -th degree Taylor polynomial for $f(x)=\ln (x)$ centred at $x=1$ . Using Taylor's Remainder Theorem, find the smallest integer $n$ so that $\left|f(x)-T_{n}(x)\right| \leq 0.0001$ for all $x \in\left[\frac{1}{2}, \frac{3}{2}\right]$ .

See Solution

Problem 293

Compute the sum of the squares of the values in the set {3, 3, 9}.

See Solution

Problem 294

Find a formula for the $n^{\text{th}}$ term $a_{n}$ of the sequence $16, 7, -2, \ldots$ .

See Solution

Problem 295

Find a formula for $a_{n}$ , the $n^{\text{th}}$ term of the sequence $36, 44, 52, \ldots$ .

See Solution

Problem 296

Find the formula for the $n^{\text{th}}$ term $a_{n}$ of the sequence: 6, 16, 26, ...

See Solution

Problem 297

Find the formula for $a_{n}$ , the $n^{\text{th}}$ term of the sequence $31, 27, 23, \ldots$

See Solution

Problem 298

Find the 70th term of the sequence $29, 17, 5, \ldots$

See Solution

Problem 299

Find the 52nd term of the sequence: $-8, -12, -16, \ldots$

See Solution

Problem 300

Find the 82nd term of the arithmetic sequence -10, 6, 22, ...

See Solution

Sequences & Series

Problem 201

12. Which explicit rule matches the sequence described in the table? \begin{tabular}{|l|c|c|c|c|} \hline n\boldsymbol{n}n & 1 & 3 & 5 & 7 \\ \hline f(n)\boldsymbol{f}(\boldsymbol{n})f(n) & -1 & -5 & -9 & -13 \\ \hline \end{tabular}

Problem 202

Problem 203

Problem 204

Find an explicit formula for the nth n^{\text {th }}nth term of this sequence. 9,−1,19,−181,..an=\begin{array}{l} 9,-1, \frac{1}{9},-\frac{1}{81}, . . \\ a_{n}= \end{array}9,−1,91​,−811​,..an​=​

Problem 205

Problem 206

En una secuencia, cada número se forma sumando los dos de arriba. ¿Qué valor tiene xxx si 26 = 10 + 16?

Problem 207

Problem 208

How much will Cardo save on the 9th day if he starts with \$75 and doubles his savings daily?

Problem 209

Cari sebutan ke-nnn, TnT_{n}Tn​, dari pola: 5=3+2(1)5=3+2(1)5=3+2(1), 9=3+3(2)9=3+3(2)9=3+3(2), 15=3+4(3)15=3+4(3)15=3+4(3).

Problem 210

Find the common ratio rrr of the geometric sequence 7, 189, yyy, 1701. Then, determine the smallest integer nnn for which the nnn-th term exceeds 10,000.

Problem 211

Solve the equation: n⋅n!−(n+1)!=−n!n \cdot n! - (n+1)! = -n!n⋅n!−(n+1)!=−n! for integer values of nnn.

Problem 212

Solve the equation (n+1)2⋅n!+(n+1)!=(n+2)!(n+1)^{2} \cdot n !+(n+1) !=(n+2) !(n+1)2⋅n!+(n+1)!=(n+2)! for n.

Problem 213

Find the 37th term of an arithmetic sequence where t1=−3t_{1}=-3t1​=−3 and t2=11t_{2}=11t2​=11.

Problem 214

Find the sum of the first 31 terms of an arithmetic series where t1=−9t_{1}=-9t1​=−9 and t2=9t_{2}=9t2​=9.

Problem 215

Rank the sums S30\mathrm{S}_{30}S30​ of these series from least to greatest: 4+15+26+…4+15+26+\ldots4+15+26+…, 77+79+81+…77+79+81+\ldots77+79+81+…, 12+17+22+…12+17+22+\ldots12+17+22+…, −178−154−130−…-178-154-130-\ldots−178−154−130−….

Problem 216

A 0.1 kg ball loses 25% of its energy when bouncing. Starting at 1 m, how many bounces until it bounces ≤ 0.25 m?

Problem 217

Lindiwe has R80 000 invested at 13.75%13.75\%13.75% p.a. and withdraws R25 000 yearly. How many years will this last?

Problem 218

Find the number of dots in the 48th figure: 4(48)+24(48)+24(48)+2. Is there a figure with 200 dots? Explain. Total dots from 10th to 30th figure: S=2m(m+1)S=2m(m+1)S=2m(m+1).

Problem 219

Parents plan to deposit \$400 on their son's 10th birthday, increasing by \$50 yearly. a. Find the deposit on his 20th birthday. b. Calculate total deposits by his 20th birthday.

Problem 220

You sold a bike for \$10,000 and invest it at 4% compounded monthly. How much can you withdraw monthly for 2.5 years?

Problem 221

Two companies produce phones. Company A starts at 600 and increases by 300 weekly. Company B starts at 500 and increases by 15%15\%15% weekly. Find: a) Phones produced in week 15 b) Total production in 15 weeks

Problem 222

A geometric sequence has first term 32 and ratio 12\frac{1}{2}21​. a. Find the first 5 terms. b. Write an explicit formula for the nnnth term of the sequence.

Problem 223

Problem 224

2ตรบกเณ์ ต (1) (ก) ศณูตาสญบูกก z=1+i+i2+i3+…+i2017z=1+i+i^{2}+i^{3}+\ldots+i^{2017}z=1+i+i2+i3+…+i2017 (8) กณกต B=1+i+i2+i3+i4+…+i100B=1+i+i^{2}+i^{3}+i^{4}+\ldots+i^{100}B=1+i+i2+i3+i4+…+i100

Problem 225

Question 11. Write the following sum as a simplified algebraic expression in ppp and qqq ∑j=37(−1)j(3pj+5q)\sum_{j=3}^{7}(-1)^{j}(3 p j+5 q)j=3∑7​(−1)j(3pj+5q)Sum === □\square□ ???

Problem 226

Question 13. Given the following arithmetic sequence 3,5,7,…3,5,7, \ldots3,5,7,…, find the value of the 14 th term and the sum of the first 28 terms. Enter your answers as integers in the appropriate boxes: (a) The 14th term is □\square□ (b) The sum of the first 28 terms is □\square□

Problem 227

Problem 228

and the inter ∑n=0∞(9x)nn!\sum_{n=0}^{\infty} \frac{(9 x)^{n}}{n!}n=0∑∞​n!(9x)n​

Problem 229

Problem 230

Problem 231

Determine the number of terms nnn in the geometr 3) 4+12+36+108…,Sn=393644+12+36+108 \ldots, S_{n}=393644+12+36+108…,Sn​=39364

Problem 232

Find each sum for the following series to the nearest tenth. Type DNE if the sum does not exist. 256+64+16+⋯S4=□S∞=□\begin{array}{l} 256+64+16+\cdots \\ S_{4}=\square \\ S_{\infty}=\square \end{array}256+64+16+⋯S4​=□S∞​=□​

Problem 233

Find the exact sum. Type your answer as a decimal, if necessary. ∑n=1∞7.4⋅0.2n−1\sum_{n=1}^{\infty} 7.4 \cdot 0.2^{n-1}n=1∑∞​7.4⋅0.2n−1

Problem 234

Problem 235

5thWork out a the nnnth term for the number of shaded squares. nnnth term =3n−2=3 n-2=3n−2 b\mathbf{b}b the n\boldsymbol{n}n th term for the number of white squares. nnnth term =n−1=n-1=n−1 ccc the nnnth term for the total number of squares. nnnth term =4n−3=4 n-3=4n−3

Problem 236

Problem 237

范 \qquad \qquad - - * \qquad - \qquad \qquad . 4,19,8,24,14,29,19,34,24,39,29,44,…4,19,8,24,14,29,19,34,24,39,29,44, \ldots4,19,8,24,14,29,19,34,24,39,29,44,… Rule: \qquad \qquad

Problem 238

Problem 239

Problem 240

Problem 241

Problem 242

7. Write Math Give an example of a rule for a pattern. List a set of numbers that fit the pattern.

Problem 243

Find the 10th and 15th terms of the sequence 3,7,11,15,19,…3, 7, 11, 15, 19, \ldots3,7,11,15,19,…

Problem 244

Find the 7th and nnnth terms of the sequence 5,3,1,−1,…5, 3, 1, -1, \ldots5,3,1,−1,…

Problem 245

Find the rrrth term of the sequence: 72, 70, 68, 66, ...

Problem 246

Find the fifteenth term of an A.P. where the 3rd term is 18 and the 7th term is 30.

Problem 247

12. Which explicit rule matches the sequence described in the table? \begin{tabular}{|l|c|c|c|c|} \hline $\boldsymbol{n}$ & 1 & 3 & 5 & 7 \\ \hline $\boldsymbol{f}(\boldsymbol{n})$ & -1 & -5 & -9 & -13 \\ \hline \end{tabular}

Find an explicit formula for the $n^{\text {th }}$ term of this sequence. $\begin{array}{l} 9,-1, \frac{1}{9},-\frac{1}{81}, . . \\ a_{n}= \end{array}$

En una secuencia, cada número se forma sumando los dos de arriba. ¿Qué valor tiene $x$ si 26 = 10 + 16?

Cari sebutan ke- $n$ , $T_{n}$ , dari pola: $5=3+2(1)$ , $9=3+3(2)$ , $15=3+4(3)$ .

Find the common ratio $r$ of the geometric sequence 7, 189, $y$ , 1701. Then, determine the smallest integer $n$ for which the $n$ -th term exceeds 10,000.

Solve the equation: $n \cdot n! - (n+1)! = -n!$ for integer values of $n$ .

Solve the equation $(n+1)^{2} \cdot n !+(n+1) !=(n+2) !$ for n.

Find the 37th term of an arithmetic sequence where $t_{1}=-3$ and $t_{2}=11$ .

Find the sum of the first 31 terms of an arithmetic series where $t_{1}=-9$ and $t_{2}=9$ .

Rank the sums $\mathrm{S}_{30}$ of these series from least to greatest: $4+15+26+\ldots$ , $77+79+81+\ldots$ , $12+17+22+\ldots$ , $-178-154-130-\ldots$ .

Lindiwe has R80 000 invested at $13.75\%$ p.a. and withdraws R25 000 yearly. How many years will this last?

Find the number of dots in the 48th figure: $4(48)+2$ . Is there a figure with 200 dots? Explain. Total dots from 10th to 30th figure: $S=2m(m+1)$ .

Parents plan to deposit \$400 on their son's 10th birthday, increasing by \$50 yearly.
a. Find the deposit on his 20th birthday. b. Calculate total deposits by his 20th birthday.

Two companies produce phones. Company A starts at 600 and increases by 300 weekly. Company B starts at 500 and increases by $15\%$ weekly. Find: a) Phones produced in week 15 b) Total production in 15 weeks

A geometric sequence has first term 32 and ratio $\frac{1}{2}$ . a. Find the first 5 terms. b. Write an explicit formula for the $n$ th term of the sequence.

2ตรบกเณ์ ต (1) (ก) ศณูตาสญบูกก $z=1+i+i^{2}+i^{3}+\ldots+i^{2017}$ (8) กณกต $B=1+i+i^{2}+i^{3}+i^{4}+\ldots+i^{100}$

Question 11. Write the following sum as a simplified algebraic expression in $p$ and $q$ $\sum_{j=3}^{7}(-1)^{j}(3 p j+5 q)$
Sum $=$ $\square$ $?$

Question 13. Given the following arithmetic sequence $3,5,7, \ldots$ , find the value of the 14 th term and the sum of the first 28 terms. Enter your answers as integers in the appropriate boxes: (a) The 14th term is $\square$ (b) The sum of the first 28 terms is $\square$

and the inter $\sum_{n=0}^{\infty} \frac{(9 x)^{n}}{n!}$

Determine the number of terms $n$ in the geometr 3) $4+12+36+108 \ldots, S_{n}=39364$

Find each sum for the following series to the nearest tenth. Type DNE if the sum does not exist. $\begin{array}{l} 256+64+16+\cdots \\ S_{4}=\square \\ S_{\infty}=\square \end{array}$

Find the exact sum. Type your answer as a decimal, if necessary. $\sum_{n=1}^{\infty} 7.4 \cdot 0.2^{n-1}$

5th
Work out a the $n$ th term for the number of shaded squares. $n$ th term $=3 n-2$ $\mathbf{b}$ the $\boldsymbol{n}$ th term for the number of white squares. $n$ th term $=n-1$ $c$ the $n$ th term for the total number of squares. $n$ th term $=4 n-3$

范 $\qquad$ $\qquad$ - - * $\qquad$ - $\qquad$ $\qquad$ . $4,19,8,24,14,29,19,34,24,39,29,44, \ldots$ Rule: $\qquad$ $\qquad$

Find the 10th and 15th terms of the sequence $3, 7, 11, 15, 19, \ldots$

Find the 7th and $n$ th terms of the sequence $5, 3, 1, -1, \ldots$

Find the $r$ th term of the sequence: 72, 70, 68, 66, ...

Find the $n$ th term of an A.P. where the 5th term is 38 and the 10th term is 23.

Find the term number of the sequence $5, 14, 23, 32, \ldots$ that equals $239$ .

Dos corredores en una pista circular se encuentran cada cuánto tiempo si uno tarda $3 \mathrm{~min}$ y el otro $5 \mathrm{~min}$ ?

Check if these sequences are bounded:
1. $a_{n}=2+3 \cdot n$
2. $a_{n}=n-1$
3. $a_{n}=1-2 \cdot n$
4. $a_{n}=1$
5. $a_{n}=-5 \cdot n$
6. $a_{n}=0$

Gib die rekursive Darstellung der Folge $x_{n}$ für $x_{n}=2^{n+1}$ , $x_{n}=2n+1$ , und $x_{n}=(-1)^{n}$ an!

Which function describes the sequence: -5, -7, -9, -11, -13, ...? A) $f(x)=-3x+2$ B) $f(x)=-2x+3$ C) $f(x)=-3x-2$ D) $f(x)=-2x-3$

Find the recursive formula for the sequence defined by $t_{n}=-5+(n-1) 78$ , with $n \in N$ and $n \geq 1$ .

Find the next 3 terms of the sequence with $a_{1} = -12$ and $a_{n} = a_{n-1} + 9$ . Choices: A) -3,-15,-27 B) -3,6,15 C) -3,-6,-3 D) -3,-6,3 E) -3,3,9

Find the next 3 terms in the sequence where $a_{n}=a_{n-1}+9$ and $a_1=-12$ . Choices: -3,-15,-27; B -3,6,15; -3,-6,-3; D -3,-6,3; -3,3,9.

Identify the function for the sequence $-4, 0, 4, 8, 12, 16, \ldots$ from the options given.

Identify the version of the sequence with an error among: 1) $f(x)=6 x+5$ , 2) $a_{1}=11$ , $a_{n+1}=a_{n}+6$ , 3) $a_{n}=11+6(n-1)$ . Fix it.

Which function represents the sequence $-4,0,4,8,12,16,\ldots$ ? A) $f(x)=4x+8$ B) $f(x)=4x-8$ C) $f(x)=-4x+8$ D) $f(x)=-4x-8$

Which function represents the sequence $3, 7, 11, 15, 19, 23, \ldots$ ? A) $f(x)=4x-4$ , B) $f(x)=4x-3$ , C) $f(x)=4x-2$ , D) $f(x)=4x-1$ .

Which TWO sequences are equivalent? A: $f(x)=7x-15$ B: $f(x)=7x-22$ C: $f(x)=7x-29$ D: $a_{1}=-22$ , $a_{n+1}=a_{n}+7$ E: $a_{1}=-29$ , $a_{n+1}=a_{n}+7$

Calculate the sum: $\sum_{n=2}^{6} \frac{n-1}{n(n+1)}$ .

Find the first three terms of a geometric sequence where the $5^{\text{th}}$ term is 1875 and the $7^{\text{th}}$ term is 46875.

Determine the common difference in the arithmetic sequence $11, 20, 29, \ldots$

Berechne das Guthaben nach 10 Jahren, wenn jährlich 5 mal $1000 €$ und 3 mal $800 €$ bei 0,7\% Zinsen eingezahlt werden.

1) Find the expression for the sequence: -66, -132, -198, -264 with $n=1$ for the first term. $a_n=$
2) Find the expression for the sequence: 7, 14, 21, 28 with $n=1$ for the first term. $a_n=$

3) Find the expression for the sequence -65, -64, -63, ... and calculate the $78^{\text{th}}$ term, $a_{78}$ .
4) Write the expression for the sequence 55, 110, ... and determine the $92^{\text{nd}}$ term, $a_{92}$ .

What is the partial sum of the arithmetic sequence where $a=17$ , $d=-3$ , and $n=47$ ?