Browse Sequences & Series problems Studdy Solved!

Problem 101

1. The sequence $a_{n}=\frac{3 n^{2}+1}{1-n^{3}}$ (A) converges to 0 . (B) converges to -1 . (C) converges to -2 . (D) is divergent.
2. The sequence $a_{n}=(-1)^{n} \frac{5^{n} 3^{n}}{7^{2 n}}$ (A) converges to 1 . (B) converges to 0 . (C) converges to 2 . (D) is divergent.
3. The sequence $a_{n}=\frac{3^{n}+1}{n+1}$ (A) converges to 1 . (B) converges to 0 . (C) converges to 2 . (D) is divergent.

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Problem 102

3. Өөр өөр насны хэдэн охид цэцэг түүцгээжээ. Түүсэн цэцгүүдээ тэд хуваахдаа хамгийн багадаа 20 цэцэг ба үлдсэний 0.04-ийг, түүний дараагийнхад 21 ба 0.04-ийг, гуравдахид 22 цэцэг ба үлдсэний 0.04-ийг гэх мэтээр хуваасаар бүгд тэнцүү авцгаав. Хэдэн охин байсан бэ? Хэчнээн цэцэг түүсэн бэ?

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Problem 103

Ten workers of equal efficiency are working on manufacturing 7 industrial widgets of different sizes. If all ten work together, it will take 13 days to make the $1^{\text {st }}$ widget, 12 days to make the $2^{\text {nd }}$ widget, 11 days to make the $3^{\text {rd }}$ widget, and so on. The factory manager decides to engage 4 workers on the $1^{\text {st }}$ widget and one worker on each of the remaining 6 widgets. No worker is removed from the work on a specific widget before it is finished. The moment the $1^{\text {st }}$ widget is complete, he assigns those 4 workers on the $2^{\text {nd }}$ widget. Once the $2^{\text {nd }}$ widget is complete, he assigns 5 workers working the $2^{\text {nd }}$ widget on the $3^{\text {rd }}$ widget, and so on. What percentage of the seventh widget was completed by the worker who started the work on that widget?
Enter your response (as an integer) using the virtual keyboard in the box provided below.

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Problem 104

Find the common difference $d$ of each arithmetic sequence. 1) $\{5,8,11,14,17, \ldots\}$ $\square$ Check Show answer 2) $\{4,0,-4,-8,-12, \ldots\}$ $\square$ Check
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Problem 105

\begin{tabular}{|c|c|c|c|c|c|c|} \hline $x$ & -2 & -1 & 0 & 1 & 2 & 3 \\ \hline $\sum_{n=0}^{\infty} a_{n}(x-b)^{n}$ & diverges & \begin{tabular}{c} converges \\ conditionally \end{tabular} & \begin{tabular}{c} converges \\ absolutely \end{tabular} & \begin{tabular}{c} converges \\ absolutely \end{tabular} & \begin{tabular}{c} converges \\ absolutely \end{tabular} & diverges \\ \hline \end{tabular}
Consider the power series $\sum_{n=0}^{\infty} a_{n}(x-b)^{n}$ , where $b$ is an integer. The convergence or divergence of the series at various values of $x$ is shown in the table above. What is the interval of convergence for the power series? (A) $(-2,2)$ (B) $(-2,3)$ (C) $[-1,2]$ (D) $[-1,3)$

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Problem 106

8 Mark for Review 40
The radius of convergence of the power series $\sum_{n=0}^{\infty} \frac{n!(n+1)!}{(2 n)!} x^{n}$ is (A) 0
B 1 (C) 2 (D) 4

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Problem 107

Use the ratio or root test to determine if the following series are convergent or divergent. If the ratio or the root test fails, indicate why.
1. $\sum_{n=1}^{\infty} \frac{1}{n!}$
5. $\sum_{n=1}^{\infty} \frac{3}{4^{n}}$
2. $\sum_{n=1}^{\infty}\left(\frac{4}{3}\right)^{n}$
6. $\sum_{n=2}^{\infty} \frac{3^{n}}{(2 n)!}$
3. $\sum_{n=1}^{\infty} \frac{1}{n+1}$
7. $\sum_{n=2}^{\infty} \frac{n}{4^{n}}$
4. $\sum_{n=1}^{\infty} \frac{(3 n)!}{n!}$
8. $\sum_{n=2}^{\infty} \frac{1}{\sqrt{n}}$

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Problem 108

The Taylor series for $\sin x$ about $x=0$ is given by $\sum_{n=1}^{\infty}(-1)^{n+1} \frac{x^{2 n-1}}{(2 n-1)!}$ and converges to $\sin x$ for all $x$ . If the ninth-degree Taylor polynomial for $\sin x$ about $x=0$ is used to approximate $\sin 2$ , what is the alternating series error bound? (A) $\frac{2^{9}}{9!}$ (B) $\frac{2^{10}}{10!}$ (C) $\frac{2^{11}}{11!}$ (D) $\frac{2^{10}}{19!}$

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Problem 109

Find the radius and interval of convergence for each of the following series.
9. $\sum_{n=2}^{\infty} \frac{(3 x-1)^{n}}{\ln (n)}$
13. $\sum_{n=1}^{\infty}(x-3)^{n}(2 n)$ !
10. $\sum_{n=1}^{\infty} \frac{(2 x-1)^{n}}{n+4}$
14. $\sum_{n=1}^{\infty}(4 x+1)^{n} n$
11. $\sum_{n=1}^{\infty} \frac{x^{n}}{(2 n)!}$
15. $\sum_{n=1}^{\infty} \frac{(x-4)^{n}}{\sqrt{n+3}}$
12. $\sum_{n=1}^{\infty} \frac{(2 x-1)^{n}}{n^{2}+1}$
16. $\sum_{n=1}^{\infty} \frac{x^{n}}{n^{\pi}}$

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Problem 110

(c) $\binom{n}{1}+2\binom{n}{2}+3\binom{n}{3}+\cdots+n\binom{n}{n}=n 2^{n-1}$ . [Hint: After expanding $n(1+b)^{n-1}$ by the binomial theorem, let $b=1$ ; note also that $\left.n\binom{n-1}{k}=(k+1)\binom{n}{k+1} \cdot\right]$

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Problem 111

Writing Recursive Formulas to Model Sequence Situations CL MATHIA < Unit Overview Step-by-Step Sample Problem Hints 88 HT G Cell Analogy to a C... C Gould Linear Functio Home Audio Support 00 System Help Finish undate Alf Bookmans Glossary Karen Nolasco I'm Done I'm Danc Marvin's football league plays its own version of the game where every offensive penalty pushes a team back half the distance to its goal line. His team is not very good, often committing many consecutive penalties. Its first penalty pushes his team back to its own 16 yard line. Another penalty puts them at their own 8 yard line, and a 3rd penalty puts them 4 yards from the goal line. A 4th penalty sets them back to the 2 yard line, and a 5th penalty puts them at the 1 yard line.
1. Recognize and Describe The first term is
2. Classify
3. Write Recursive Formula Each term is equal to the previous term Assuming the team commits one more consecutive penalty, how many yards will it be from its goal line? A CARNEGIE © 2023 Carnegie Learning LEAN 1:20 PM 11/17/2024

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Problem 112

< Unit Overview Step-by-Step Sample Problem Hints
Norman discovers a new animal species that has a lifespan of one week and has 2 offspring just before dying. At the end of the 1 st week, Norman counts 3 such animals. A week later animals, and a week after that there are 12 animals. After the 4 th week, there are 24 animals, and a week later there are 48 animals.
1. Recognize and Describe
The first term is 3 . Each term is equal to the previous term $\square$ $\square$ .
2. Classify
3. Write Recursive Formula
How many animals would there be after another week? $\square$

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Problem 113

Verify using the indicated test that the infinite series converges. (Hint: Use partial fractions.) $\sum_{n=1}^{\infty} \frac{1}{n(n+1)}$
By the telescoping series test, we have $\sum_{n=1}^{\infty} \frac{1}{n(n+1)}=\lim _{n \rightarrow \infty}$ $\square$ ) $=$ $\square$ , and thus the series converges.

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Problem 114

Determine whether the given series is convergent or divergent. $\sum_{n=1}^{\infty} \frac{n}{\sqrt{2 n^{2}+1}}$ convergent divergent

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Problem 115

17. [-/1 Points] DETAILS MY NOTES
Find a power series representation for the function. (Give your power series representation centered at $\mathrm{x}=0$ .) $\begin{array}{c} f(x)=\frac{x}{4+x^{2}} \\ f(x)=\sum_{n=0}^{\infty} \end{array}$
Determine the interval of convergence. (Enter your answer using interval notation.) $\square$

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Problem 116

Decide whether the following statement makes sense (or is clearly true) or does not make sense (or is clearly false). Explain your reasoning. If interest rates stay at $6 \%$ APR and I continue to make my monthly $\$ 50$ deposits into my retirement plan, I should have at least $\$ 40,000$ saved when I retire in 35 years.
The statement $\square$ does not make sense because I will have \\square$ in my retirement account when I retire in 35 years. (Round to the nearest cent as needed.)

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Problem 117

For the following function, find the Taylor series centered at $x=\frac{\pi}{2}$ and then give the first 5 nonzero terms of the Taylor series and the open interval of convergence. $f(x)=\cos (x)$ $f(x)=\sum_{n=0}^{\infty}$ $\square$ $\square$ $\square$ $+$ $\square$ $f(x)=\square 0_{0}+\square+\square a_{0}+\square$ $\square$ $+\cdots$
The open interval of convergence is: (-infinity, infinity) $\square$ (Give your answer in interval notation.)
Submit answer Next item

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Problem 118

8. Find the indicated arithmetic term. a) $a=5, d=3$ ; find $t_{12}$ c) $a=-\frac{3}{4}, d=\frac{1}{2}$ ; find $t_{10}$ e) $a=-0.75, d=0.05$ ; find $t_{40}$

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Problem 119

Find the nth , or general, term for the geometric sequence. $2,-2,2,-2, \ldots$
Write an expression for the nth term.

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Problem 120

Ex: $\sum_{n=1}^{\infty} \frac{(x-3)^{n}}{5^{n}(n+1)}$

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Problem 121

3 It is given that, at any point on the graph of $y=\mathrm{f}(x), \frac{\mathrm{d} y}{\mathrm{~d} x}=\mathrm{e}^{x} \sin x-y$ . (i) Show that $\frac{\mathrm{d}^{3} y}{\mathrm{~d} x^{3}}=\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}-2 y$ .
The graph of $y=\mathrm{f}(x)$ passes through the origin $O$ . (ii) Find the Maclaurin series for $y$ , up to and including the term in $x^{4}$ . [4] (iii) Hence, find the Maclaurin series for $\mathrm{e}^{2 x} \sin 2 x$ , up to and including the term in $x^{2}$ . [2]

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Problem 122

\begin{tabular}{|c|c|c|} \hline $\begin{array}{lll} A .2155 & B .2000 & C .2144 \end{array}$ & $\text { D. } 1800$ & $\text { E. } 1758$ \\ \hline 6.032. Хэрвээ $a_{1}=7 ; d=15$ бол $\sum_{k=7}^{15} a_{k}-$ г ол. & & \\ \hline A. 1680 B. $300 \quad$ C. 315 & D. 1413 & E. 1331 \\ \hline $\begin{array}{r} \text { 6.033. Хэрвээ } a_{1}=8 ; a_{2}=8,5 \text { бол } \sum_{k=3}^{18} a_{k}-\text { г ол. } \\ \text { A. } 204 \quad \text { B. } 205 \quad \text { C. } 208 \end{array}$ & D. 180 & E. 187 \\ \hline \begin{tabular}{l} 6.034. Хэрвээ $a_{2}=3 ; a_{3}=-1$ бол $\sum_{k=14}^{29} a_{k}-$ г ол. \\ A. -1100 \\ B. -1150 \\ C. -1200 \end{tabular} & D. -1350 & E. -1254 \\ \hline $\begin{array}{rlrl} \text { 6.035. Хэрвээ } a_{1}=1 ; a_{3}=4 & \text { бол } \sum_{k=20}^{31} a_{k}-\text { г ол. } \\ & A .435,7 & \text { B. } 453 & C .574,4 \end{array}$ & D.387,4 & E.400,8 \\ \hline $\begin{array}{ccc} \text { 6.036. Хэрвээ } a_{1}=2 ; d=-0,5 \text { бол } \sum_{k=13}^{31} a_{k}-\text { г ол. } \\ A .-1480,7 & \text { B. }-180 & \text { C. }-161,5 \end{array}$ & D. $-149,5$ & 5 E. -150 \\ \hline $\begin{array}{r} \text { 6.037. Хэрвээ } a_{3}=1 ; a_{4}=-0,5 \text { бол } \sum_{k=10}^{24} a_{k}-\text { г ол. } \\ \text { A. }-300 \\ \text { B. }-200 \\ \text { C. }-245 \end{array}$ & D. -287 & E. -310 \\ \hline $\begin{array}{rrr} \text { 6.038. Хэрвээ } a_{1}=-2 ; d=0,5 \text { бол } \sum_{k=18}^{30} a_{k}-\text { г ол. } \\ A .119,5 & \text { B. } 119,75 & \text { C. } 120 \end{array}$ & D.123,5 & E.120,25 \\ \hline $\begin{array}{l} \text { 6.039. Хэрвээ } a_{1}=-1 ; d=-0,2 \text { бол } \sum_{k=31}^{40} a_{k}-\text { гол. } \\ \begin{array}{lll} \text { A. }-81 & \text { B. }-79 & \text { C. }-80 \end{array} \end{array}$ & D. 79 & E. 80 \\ \hline $\begin{aligned} \text { 6.040. Хэрвээ } a_{1}= & 0,2 ; d=0,4 \text { бол } \sum_{k=150}^{171} a_{k}-\text { г ол. } \\ \text { A. } 1400 & \text { B. } 1406 \quad \text { C. } 1409 \end{aligned}$ & D. 1408 & E. 1405 \\ \hline \end{tabular}

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Problem 123

6.045. Арифметик прогрессийн 5 - р гишүүн нь 3 - р гишүүнээс 3 - аар их ба тэдгээрийн нийлбэр нь 10 бол $2-\mathrm{p}$ гишүүнийг ол. A. 2,4 B. 4 C. 8 D. 2 E. 8,4 6.046. Арифметик прогрессийн гурав ба зургаадугаар гишүүдийн нийлбэр 3,5 . Эхний 8 гишүүний нийлбэр ол. A. 1 B. 14 C. 12 D. 15 E. 2 6.047. Арифметик прогрессийн 6 - р гишүүн нь 4 - р гишүүнээсээ 8 - аар их ба тэдгээрийн нийлбэр 33 бол $3-\mathrm{p}$ гишүүнийг ол. A. 7,5 B. 9,5 C. 8,5 D. 5 E. 5,5

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Problem 124

6.073. $71+67+63+\cdots-53$ A. 288 B. 284 C. 280 D. 302 E. 306 6.074. $1+1 \frac{1}{6}+1 \frac{1}{3}+\cdots+4 \frac{1}{2}$ A. $60 \frac{2}{3}$ B. $60 \frac{1}{3}$ C. 60,5 D. 70 E. 70,5 \begin{tabular}{|c|c|c|c|} \hline \multicolumn{4}{|l|}{6.075. $2,01+2,02+2,03+\cdots+3,00$ } \\ \hline A.250,5 B.250,49 & C. 250,51 & D. 250,61 & E. 250,62 \\ \hline \multicolumn{4}{|l|}{6.076. $-10-7-4-\cdots+50$ } \\ \hline A. 423 B. 417 & C. 426 & D. 420 & E. 429 \\ \hline \multicolumn{4}{|l|}{6.077. $2,7+3,2+3,7+\cdots+17,7$ } \\ \hline A.316,2 B.316,7 & C. 317,2 & D.315,8 & E.315,3 \\ \hline \multicolumn{4}{|l|}{6.078.407 + $401+395+\cdots-133$ } \\ \hline A. $12461 \quad$ B. 12471 & C. 12467 & D. 12481 & E. 12487 \\ \hline \multicolumn{4}{|l|}{6.079. $50+47+44+\cdots+14$ } \\ \hline A. 416 B. 419 & C. 422 & D. 413 & E. 410 \\ \hline \multicolumn{4}{|l|}{ $6.080 .53+50+47+\cdots-4$ } \\ \hline A. 487 B. 484 & C. 490 & D. 493 & E. 496 \\ \hline \end{tabular}

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Problem 125

8. (a) Verify that for all $n \geq 1$ , $2 \cdot 6 \cdot 10 \cdot 14 \cdots(4 n-2)=\frac{(2 n)!}{n!}$ (b) Use part (a) to obtain the inequality $2^{n}(n!)^{2} \leq(2 n)$ ! for all $n \geq 1$ .

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Problem 126

Carey is organizing her books and putting them on shelves. She put 2 books on the first shelf, 8 books on the second shelf, 32 books on the third shelf, and 128 books on the fourth shelf. What kind of sequence is this?

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Problem 127

8. (a) Verify that for all $n \geq 1$ , $2 \cdot 6 \cdot 10 \cdot 14 \cdots \cdots(4 n-2)=\frac{(2 n)!}{n!}$ (b) Use part (a) to obtain the inequality $2^{n}(n!)^{2} \leq(2 n)$ ! for all $n \geq 1$ .

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Problem 128

Agebraic Thinking: Look at the patiern below. $8,21,34,47,60$
What could be a rule for this patfern? A) Add 13 B) Add 15 C) Add II - $)$ Subtract 13

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Problem 129

Find the first six terms of the piecewise sequence: $\begin{array}{l} a_{n}=\left\{\begin{array}{ll} \frac{n^{2}}{2 n+3} & \text { if } n \leq 4 \\ n^{2}-2 & \text { if } n>4 \end{array}\right. \\ a_{1}=\square \\ a_{2}=\square \\ a_{3}=\square \\ a_{4}=\square \\ a_{5}=\square \\ a_{6}=\square \end{array}$

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Problem 130

Write an explicit formula for the sequence: $\begin{array}{l} \left\{\frac{4}{3}, \frac{6-e^{1}}{4+e^{1}}, \frac{7-e^{2}}{6+e^{2}}, \frac{8-e^{3}}{8+e^{3}}, \frac{9-e^{4}}{10+e^{4}}, \frac{10-e^{5}}{12+e^{5}}, \cdots\right\} \\ a_{n}= \end{array}$

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Problem 131

Question Watch
Find the common ratio of the geometric sequence $20,160,1280, \ldots$
Answer Attempt 2 out of 2 $\square$ Submit Answer

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Problem 132

Write the arithmetic sequence $-17,-13,-9,-5, \ldots$ in the standard form: $a_{n}=$ $\square$

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Problem 133

Write the arithmetic sequence $-3,-10,-17,-24, \ldots$ in the standard form: $a_{n}=$ $\square$

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Problem 134

Determine the Convergence or divergence of these series: (1) $\sum_{n=0}^{\infty} \frac{(3+i)^{2 n}}{(2 n)!}$ (2) $\sum_{n=0}^{\infty} n\left(\frac{i}{2}\right)^{n}$ (3) $\sum_{n=1}^{\infty} \frac{i^{n}}{2^{n}}$

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Problem 135

Find the radius and center of the Convergenced of these scries: (1) $\sum_{n=0}^{\infty}(z-2 i)^{n}$ (5) $\sum_{n=0}^{\infty} \frac{(-1)^{n} z^{n}}{n!}$ (2) $\sum_{n=0}^{\infty} 6^{n}(z-i)^{n}$ (6) $\sum_{n=0}^{\infty} n\left(\frac{z}{3}\right)^{n}$ (3) $\sum_{n=0}^{\infty}(n!)^{2} z^{n}$ (4) $\sum_{n=1}^{\infty} \frac{z^{2 n}}{n!}$

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Problem 136

check the Cenvergence of the series. then calculate the radius of Convergence. Find the radius of convergence using derivatives and integration by parts. (1) $\sum_{n=0}^{\infty} \frac{z^{n}}{n!}$ (2) $\sum_{n=0}^{\infty} z^{n}$

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Problem 137

BIG IDEAS MATH CHLOE ROGERS \#17 i Listen
Write an equation for the $n$ th term of the arithmetic sequence. Then find $a_{25}$ . $51,48,45,42, \ldots$ $\begin{array}{l} a_{n}=\square \\ a_{25}=\square \end{array}$
Previous 12 13 14 15 16 17 18 19 20 21 Next

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Problem 138

Calculate the series: $\sum_{n=1}^{\infty} n\left(\frac{i}{1-i}\right)^{n-1}$

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Problem 139

A sequence can be generated by using $a_{n}=4+(n-1)$ , where $n$ is a whole number greater than 1 . What are the first four terms of the sequence? A. $4,11,18,25$ B. $4,-3,-10,-17$ $a_{2}=4+(2-1)$ C. $4,8,12,16$ $a_{2}=4+1$ D. $4,-1,-6,-11$ $a_{2}=5$

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Problem 140

A section in a stadium has 24 seats in the first row, 28 seats in the second row, increasing by 4 seats each row for a total of 32 rows. How many seats are in this section of the stadium? $\square$ seats

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Problem 141

7. Since being adopted as the national symbol for the United States in 1782 , the bald eagle population has experienced large fluctuations. By the early 1960's, the bald eagle nearly became extinct. In 1963 , there were only 417 nesting pairs of bald eagles in the US. The population of bald eagles can be modeled using a geometric sequence where 1963 represents year 0. In 2019 (year 56), there were a total of 71,400 nesting pairs of bald eagles across the US. Based on this model, how many nesting pairs of bald eagles were in the US in 2000 ?

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Problem 142

Directions: Let $a_{n}$ be an arithmetic sequence with the following properties. For each of the following, find an expression for $a_{n}$ , and then find $a_{11}$ .
7. $a_{3}=7$ and $a_{8}=17$
8. $a_{2}=-3$ and $a_{6}=-9$
9. $a_{5}=7$ and $d=-4$
10. $a_{4}=-1$ and $d=\frac{2}{3}$ 12.

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Problem 143

write the taylor expansion of function: $f(z)=\frac{1}{3+z}$ around the Point: $z_{0}=2 i$

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Problem 144

Consider the following series. $\sum_{n=1}^{\infty} 6^{n+1} 7^{-n}$
Determine whether the geometric series is convergent or divergent. Justify your answer. Converges; the series is a constant multiple of a geometric series. Converges; the limit of the terms, $a_{n^{\prime}}$ is 0 as $n$ goes to infinity. Diverges; the limit of the terms, $a_{n^{\prime}}$ is not 0 as $n$ goes to infinity. Diverges; the series is a constant multiple of the harmonic series.
If it is convergent, find the sum. (If the quantity diverges, enter DIVERGES.)

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Problem 145

Discuter, suivant : Avec des paramètres 1 .
1. $u_{n}=e^{\frac{1}{n}-a-\frac{b}{n}}, \quad a, b \in \mathbb{R}$
2. $u_{n}=\cos \left(\frac{1}{n}\right)-a-\frac{b}{n}, \quad a, b \in \mathbb{R}$
3. $u_{n}=\frac{1}{a n+b-\frac{c}{n}}, \quad a, b, c \in \mathbb{R}, \quad(a, b) \neq(0,0)$

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Problem 146

Use the Ratio Test to determine whether the series is convergent or divergent. $\sum_{n=1}^{\infty} \frac{n \pi^{n}}{(-5)^{n-1}}$
Identify $a_{n}$ . $\square$
Evaluate the following limit. $\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|$ $\square$
Since $\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|$ ? 1, ---Select--- $\square$

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Problem 147

Use the Root Test to determine whether the series convergent or divergent. $\sum_{n=2}^{\infty}\left(\frac{-7 n}{n+1}\right)^{3 n}$
Identify $a_{n}$ . $\square$
Evaluate the following limit. $\lim _{n \rightarrow \infty} \sqrt[n]{\left|a_{n}\right|}$ $\square$
Since $\lim _{n \rightarrow \infty} \sqrt[n]{\left|a_{n}\right|}$ $\square$ ? 1, $\square$ -Select--

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Problem 148

4. If the sum is 220 and the first term is 10 , find the common difference if the last term is 30. A. 2 B. 5 C. 3 D. $2 / 3$

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Problem 149

5. Determine the sum of the progression if there are 7 arithmetic mean between 3 and 35.
171 B. 182 C. 232 D. 216

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Problem 150

10. Do the following series converge or diverge? Be sure to state which test you are using show that that test is able to be applied. All work must be shown. (a) $\sum_{n=1}^{\infty} \frac{1}{(2 n+7)^{3}}$ (b) $\sum_{n=1}^{\infty} \frac{(n-1)!}{5^{n}}$ (c) $\sum_{n=1}^{\infty} \frac{1}{n \sin ^{2}(n)}$ (d) $\sum_{n=1}^{\infty} \frac{n^{3}-2 n^{2}+n-1}{n^{5}-2}$ (e) $\sum_{n=1}^{\infty} \frac{(-1)^{n} n}{\sqrt{n+1}}$ (f) $\sum_{n=1}^{\infty} \frac{n^{2 n}}{\left(1+2 n^{2}\right)^{n}}$ (g) $\sum_{n=1}^{\infty} \frac{(-2)^{1+3 n}(n+1)}{n^{2} 5^{1+n}}$ (h) $\sum_{n=1}^{\infty} \ln \left(\frac{n}{3 n+1}\right)$ (i) $\sum_{n=1}^{\infty} \frac{e^{1 / n}}{n^{2}}$ (j) $\sum_{n=1}^{\infty}(-1)^{n+1} \frac{n^{2}}{n^{3}+1}$

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Problem 151

Write an explicit formula for $a_{n}$ , the $n^{\text {th }}$ term of the sequence $15,7,-1, \ldots$

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Problem 152

Write an explicit formula for $a_{n}$ , the $n^{\text {th }}$ term of the sequence $8,-32,128, \ldots$ .

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Problem 153

Chapter $1^{*}$ $\qquad$ 28 points / ALL WORK FOR CREDIT! Find the sum of the first 42 terms in the following sequence: $11,17,23, \ldots$

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Problem 154

Algebra Quiz 9.4-9.6 Exponentials \& Sequences
Question 1 of 6
Question 1
What is the equation for the $n$ th term of the geometric sequence $2,8,32,128, \ldots$ A) $a_{n}=2(6)^{n-1}$ B) $a_{n}=2(4)^{n-1}$ C) $a_{n}=8^{n-1}$ D) $a_{n}=2+4^{n-1}$

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Problem 155

$\sum_{n=1}^{\infty} \frac{2}{\sqrt{n}+2}$

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Problem 156

5. Determine whether the series $\sum_{k=1}^{\infty} \frac{7^{k-2}}{8^{k+1}}$ converges or diverges. If it converges, find its sum. (A) converges and equals $\frac{1}{64}$ (B) converges and equals $\frac{1}{56}$ (C) converges and equals $\frac{1}{8}$ (D) The series diverges.

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Problem 157

(11) These numbers make up two linear sequences.
1 3 4 5 7 7 10 13 What are the two linear sequences? $\qquad$ $\qquad$ $\qquad$ $2^{\text {nd }}$ $\qquad$ $\qquad$ $\qquad$ $\qquad$

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Problem 158

reach series will converge or diverge. Then sible. 7) $8.1+3.24+1.296+0.5184 \ldots$

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Problem 159

Determine whether the series $\sum_{k=0}^{\infty} \frac{k}{\sqrt{k^{2}+3}}$ converges.

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Problem 160

Evaluate the infinite geometric series: $1+1 / 5+1 / 25+\ldots$
1 2 3
65/27 5/4 $9 / 5$ 5/6

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Problem 161

1) What is the series of the sequence $\{1,4,7,10,13\}$ ? $\square$ Check Show answer 2) What is $a_{10}$ for the sequence $\{2,6,10, \ldots\}$ ? $\square$ Check Show answer 3) What is $S_{10}$ for the sequence $\{2,6,10, \ldots\}$ ? $\square$ Check Show answer. 4) What is $\sum_{n=1}^{20} 3 n$ ? $\square$ Check Show answer

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Problem 162

What are the first three terms of the arithmetic sequence $a_{n}=-4-7(n-1)$ ? $\begin{array}{l} a_{1}=\text { Ex:10 } 0 \\ a_{2}=\square \\ a_{3}=\square \end{array}$

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Problem 163

witite the firist tive
13. $a_{1}=2 ; a_{n}=a_{n-1}+5$
14. $a_{1}=-1 ; a_{n}=a_{n-1}+8$ $16 . a_{1}=6 ; a_{n}=a_{n-1}-5$
17. $a_{1}=-1 ; a_{n}=a_{n-1}+5$

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Problem 164

27-28 Explain why the Integral Test can't be used to determine whether the series is convergent.
27. $\sum_{n=1}^{\infty} \frac{\cos \pi n}{\sqrt{n}}$
28. $\sum_{n=1}^{\infty} \frac{\cos ^{2} n}{1+n^{2}}$

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Problem 165

2. (a) What is a convergent sequence? Give two examples. (b) What is a divergent sequence? Give two examples.

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Problem 166

Question Watch Video Show Examples
What is a formula for the nth term of the given sequence? $36,24,16 \ldots$
Answer $a_{n}=36\left(\frac{3}{2}\right)^{-n}$ $a_{n}=54\left(\frac{3}{2}\right)^{1-n}$ Submit Answer $a_{n}=36\left(\frac{2}{3}\right)^{n-1}$ $a_{n}=54\left(\frac{2}{3}\right)^{n-1}$ Copyright 92024 DéltaMath.com All Rights Reserved. Privacy Policy I Terms of Service $55^{\circ} \mathrm{F}$ Mostly cloudy 532 PM 11/19/2024

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Problem 167

What is a formula for the nth term of the given sequence? $20,23,26 \ldots$
Answer
$\begin{array}{l} a_{n}=20+3(n+1) \\ a_{n}=23-3 n \\ a_{n}=20+3(n-1) \\ a_{n}=20(3)^{n} \end{array}$ Submit Answer $55^{\circ} \mathrm{F}$ Mostly cloudy (4) $\begin{array}{c}533 \mathrm{PM} \\ 11 / 19 / 2024\end{array}$

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Problem 168

Question Watch Video Show Examples
Write an explicit formula for $a_{n}$ , the $n^{\text {th }}$ term of the sequence $2,-10,50, \ldots$
Answer Attempt 2 out of 2 $a_{n}=$ $\square$ Submit Answer Still Stuck?

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Problem 169

I'm sorry, but I can't assist with that request.

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Problem 170

57-63 Calcule los valores de $x$ para los cuales la serie converge. Determine la suma de la serie para dichos valores de $x$ .
57. $\sum_{n=1}^{\infty}(-5)^{n} x^{n}$
58. $\sum_{n=1}^{\infty}(x+2)^{n}$

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Problem 171

2-30 Determine si la serie es absolutamente convergente, condicionalmente convergente o divergente.
3. $\sum_{n=1}^{\infty} \frac{n}{5^{n}}$
4. $\sum_{n=1}^{\infty}(-1)^{n-1} \frac{n}{n^{2}+4}$

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Problem 172

Find the eighth term of the geometric sequence, given the first term and common ratio. $\begin{array}{c} a_{1}=6 \text { and } r=-1 / 3 \\ -\frac{[?]}{[\quad} \end{array}$

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Problem 173

Find the fourth term of the geometric sequence, given the first term and common ratio. $a_{1}=8 \text { and } r=4$ $\square$

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Problem 174

Find the final amount in the following retirement account, in which the rate of return on the account and the regular contribution change over time. $\$ 228$ per month invested at $6 \%$ , compounded monthly, for 7 years; then $\$ 577$ per month invested at $7 \%$ , compounded monthly, for 7 years.
What is the amount in the account after 14 years? \$ (Do not round until the final answer. Then round to the nearest dollar as needed.)

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Problem 175

Find the Taylor series about 0 for each of the functions below. Give the first three nonzero terms for each. A. $x^{2} \sin \left(x^{2}\right)-x^{4}=$ $\square$ $+$ $\square$ $\square$ $+$ $+\cdots$ B. $2 \cos (x)+x^{2}=$ $\square$ $+$ $\square$ $+$ $\square$ $+\cdots$
For each of these series, also be sure that you can find the general term in the series!

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Problem 176

In a certain simulation, the population of a bacteria colony can be modeled using a geometric aequence, where the first day of the simulation is day 1 . The population on day 4 was 4,000 bacteria, and the population on day 8 was 49,000 bacteria. What was the population of the colony on day 6 based on the simulation? (A) 26,500 (B) 26,192 (C) 14,000 (D) 611

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Problem 177

Find the sum. $\quad \sum_{n=3}^{7} 2^{n}$

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Problem 178

Question Find the 11 th term of the geometric sequence $2,-6,18, \ldots$ Answer Attempt 1 out of 2

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Problem 179

Determine if each infinite geometric series diverges or converges. If the series converges, type the sum. If the series diverges, type: diverges 1) $\sum_{i=1}^{\infty}\left(3 \cdot\left(\frac{1}{4}\right)^{i-1}\right)$ $\square$ Check Show answer 2) $\sum_{i=1}^{\infty}\left(-2 \cdot 3^{i-1}\right)$ $\square$ Check Show answer 3) $\sum_{i=1}^{\infty}\left(4 \cdot 1^{i-1}\right)$
Check Show answer 4) $\sum_{i=1}^{\infty}\left(-10 \cdot\left(-\frac{1}{4}\right)^{i-1}\right)$

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Problem 180

(2) $\sum_{k=1}^{\infty} \frac{\sqrt{2 k+1}}{3 k^{3}+3 k+1}$

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Problem 181

What are the first three terms of the geometric sequence $a_{n}=-7 \cdot 3^{n-1}$ ? $\begin{array}{l} a_{1}=\operatorname{Ex} 123 \hat{2} \\ a_{2}=\operatorname{Ex} 123 \hat{2} \\ a_{3}=E x \cdot 123 \hat{2} \end{array}$

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Problem 182

Write the first four terms of the sequence whose general term is given below. $\begin{array}{l} a_{n}=\frac{n^{4}}{(n+1)!} \\ a_{1}=\frac{1}{2} \end{array}$ (Type an integer or a simplified fraction.) $\mathrm{a}_{2}=\square$ (Type an integer or a simplified fraction.)

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Problem 183

Drill Problem 5.7 Consider a continuous-time signal defined by $g(t)=\frac{\sin (\pi t)}{\pi t}$
The signal $g(t)$ is uniformly sampled to produce the infinite sequence $\left\{g\left(n T_{s}\right)\right\}_{n=-\infty}^{\infty}$ . Determine the condition that the sampling period $T_{s}$ must satisfy so that the signal $g(t)$ is uniquely recovered from the sequence $\left\{g\left(n T_{s}\right)\right\}$ .

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Problem 184

Write the explicit rule for the geometric sequence $3,15,75,375,1875$ .

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Problem 185

The 15 th term of the arithmetic sequence is 33 and the 50 th term is 103 . What is the 79th partial sum of the arithmetic sequence?

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Problem 186

$3 / 7$
Find the sum of the 1 st 112 terms of: $7+11$ $+15+\ldots .$

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Problem 187

A grocer is creating a display of cans. There are 40 cans on the 1st (bottom) row, 38 cans on the 2 nd row, 36 cans on the 3 rd row, and so on. How many cans will be on the 15th

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Problem 188

$Q_{3}$ -The sequence $\{\ln (n)\}_{n=5}^{\infty}$ is: (a) not bounded above b) bounded above by $(\ln 5)$ c) bounded below by (5) d) none

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Problem 189

The first three terms of a sequence are given. Write your answer as a decimal or whole number. Round to the nearest thousandth (if necessary). $255,250,245, \ldots$
Find the 40 th term.

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Problem 190

The first three terms of a sequence are given. Write your answer as a decimal or whole number. Round to the nearest thousandth (if necessary). $2,7,12, \ldots$
Find the 46 th term.

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Problem 191

The first three terms of a sequence are given. Write your answer as a decimal or whole number. Round to the nearest thousandth (if necessary). $9,15,21, \ldots$
Find the 38 th term.

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Problem 192

Find the 76 th term of the arithmetic sequence $16,14,12, \ldots$

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Problem 193

15. $f(x)=\sum_{n=1}^{\infty} \frac{1 \cdot 3 \cdot 5 \cdots(2 n-1)}{3 \cdot 6 \cdot 9 \cdots(3 n)} x^{n}$

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Problem 194

A new shopping mall records 150 total shoppers on their first day of business. Each day after that, the number of shoppers is $15 \%$ more than the number of shoppers the day before.
Which expression gives the total number of shoppers in the first $n$ days of business?
Choose 1 answer: (A) $1.15\left(\frac{1-150^{n}}{1-150}\right)$ (B) $0.85\left(\frac{1-150^{n}}{1-150}\right)$ (C) $150\left(\frac{1-1.15^{n}}{1-1.15}\right)$ (D) $150\left(\frac{1-0.85^{n}}{1-0.85}\right)$

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Problem 195

12. Given a geometric sequence whose $n^{t h}$ term is $g_{n}=-2(81)^{\frac{-n}{4}}$ , are the terms of this sequence increasing or decreasing? Explain.

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Problem 196

Задание 3. Мистер Джонсон по случаю своего тридцатилетия открыл 1 октября 2010 года в банке счёт, на который он ежегодно кладет 6000 рублей. По условиям вклада банк ежегодно начисляет $30 \%$ на сумму, находящуюся на счёте. Через 7 лет 1 октября 2017 года октября, следуя примеру мистера Джонсона, мистер Браун по случаю своего тридцатилетия тоже открыл в банке счет, на который ежегодно кладёт по 13800 рублей, а банк начисляет $69 \%$ в год. В каком году после очередного пополнения суммы вкладов мистера Джонсона и мистера Брауна сравняются, если деньги со счетов не снимают?

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Problem 197

Montrer par riecurence $\left(\forall n \in \mathbb{N}^{*}\right)$ $1+3+5+\cdots+(2 n-1)=n^{2}$

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Problem 198

51-56 - Partial Sums of an Arithmetic Sequence Find the partial sum $S_{n}$ of the arithmetic sequence that satisfies the given conditions.
51. $a=3, \quad d=5, \quad n=20$

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Problem 199

Write the first four elements of the sequence. 1) $\frac{n+1}{3 n-1}$

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Problem 200

Pattern A and Pattern B both start at 0 . Which statement about the patterns is true? \begin{tabular}{|l|c|c|c|c|c|c|} \hline Pattern A & 0 & 6 & 12 & 18 & 24 & 30 \\ \hline Pattern B & 0 & 2 & 4 & 6 & 8 & 10 \\ \hline \end{tabular}
Each term in Pattern $A$ is 4 more than its corresponding term in Pattern B.
Each term in Pattern A is 3 times its corresponding term in Pattern B.
Each term in Pattern $A$ is 6 times its corresponding term in Pattern B.
Each term in Pattern A is 8 more than its corresponding term in Pattern B.

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Sequences & Series

Problem 101

Problem 102

Problem 103

Problem 104

Find the common difference ddd of each arithmetic sequence. 1) {5,8,11,14,17,…}\{5,8,11,14,17, \ldots\}{5,8,11,14,17,…} □\square□ Check Show answer 2) {4,0,−4,−8,−12,…}\{4,0,-4,-8,-12, \ldots\}{4,0,−4,−8,−12,…} □\square□ CheckShow answer

Problem 105

Problem 106

8 Mark for Review 40The radius of convergence of the power series ∑n=0∞n!(n+1)!(2n)!xn\sum_{n=0}^{\infty} \frac{n!(n+1)!}{(2 n)!} x^{n}∑n=0∞​(2n)!n!(n+1)!​xn is (A) 0B 1 (C) 2 (D) 4

Problem 107

Problem 108

Problem 109

Problem 110

Problem 111

Problem 112

Problem 113

Problem 114

Determine whether the given series is convergent or divergent. ∑n=1∞n2n2+1\sum_{n=1}^{\infty} \frac{n}{\sqrt{2 n^{2}+1}}n=1∑∞​2n2+1​n​ convergent divergent

Problem 115

Problem 116

Problem 117

Problem 118

8. Find the indicated arithmetic term. a) a=5,d=3a=5, d=3a=5,d=3; find t12t_{12}t12​ c) a=−34,d=12a=-\frac{3}{4}, d=\frac{1}{2}a=−43​,d=21​; find t10t_{10}t10​ e) a=−0.75,d=0.05a=-0.75, d=0.05a=−0.75,d=0.05; find t40t_{40}t40​

Problem 119

Find the nth , or general, term for the geometric sequence. 2,−2,2,−2,…2,-2,2,-2, \ldots2,−2,2,−2,…Write an expression for the nth term.

Problem 120

Ex: ∑n=1∞(x−3)n5n(n+1)\sum_{n=1}^{\infty} \frac{(x-3)^{n}}{5^{n}(n+1)}∑n=1∞​5n(n+1)(x−3)n​

Problem 121

Problem 122

Problem 123

Problem 124

Problem 125

Problem 126

Carey is organizing her books and putting them on shelves. She put 2 books on the first shelf, 8 books on the second shelf, 32 books on the third shelf, and 128 books on the fourth shelf. What kind of sequence is this?

Problem 127

Problem 128

Agebraic Thinking: Look at the patiern below. 8,21,34,47,608,21,34,47,608,21,34,47,60What could be a rule for this patfern? A) Add 13 B) Add 15 C) Add II - ))) Subtract 13

Problem 129

Problem 130

Problem 131

Question WatchFind the common ratio of the geometric sequence 20,160,1280,…20,160,1280, \ldots20,160,1280,…Answer Attempt 2 out of 2 □\square□ Submit Answer

Problem 132

Write the arithmetic sequence −17,−13,−9,−5,…-17,-13,-9,-5, \ldots−17,−13,−9,−5,… in the standard form: an=a_{n}=an​= □\square□

Problem 133

Write the arithmetic sequence −3,−10,−17,−24,…-3,-10,-17,-24, \ldots−3,−10,−17,−24,… in the standard form: an=a_{n}=an​= □\square□

Problem 134

Problem 135

Problem 136

check the Cenvergence of the series. then calculate the radius of Convergence. Find the radius of convergence using derivatives and integration by parts. (1) ∑n=0∞znn!\sum_{n=0}^{\infty} \frac{z^{n}}{n!}∑n=0∞​n!zn​ (2) ∑n=0∞zn\sum_{n=0}^{\infty} z^{n}∑n=0∞​zn

Problem 137

Problem 138

Calculate the series: ∑n=1∞n(i1−i)n−1\sum_{n=1}^{\infty} n\left(\frac{i}{1-i}\right)^{n-1}n=1∑∞​n(1−ii​)n−1

Problem 139

Problem 140

A section in a stadium has 24 seats in the first row, 28 seats in the second row, increasing by 4 seats each row for a total of 32 rows. How many seats are in this section of the stadium? □\square□ seats

Problem 141

Problem 142

Problem 143

write the taylor expansion of function: f(z)=13+zf(z)=\frac{1}{3+z}f(z)=3+z1​ around the Point: z0=2iz_{0}=2 iz0​=2i

Problem 144

Problem 145

Problem 146

Problem 147

Problem 148

4. If the sum is 220 and the first term is 10 , find the common difference if the last term is 30. A. 2 B. 5 C. 3 D. 2/32 / 32/3

Problem 149

5. Determine the sum of the progression if there are 7 arithmetic mean between 3 and 35.171 B. 182 C. 232 D. 216

Problem 150

Problem 151

Write an explicit formula for ana_{n}an​, the nth n^{\text {th }}nth term of the sequence 15,7,−1,…15,7,-1, \ldots15,7,−1,…

Problem 152

Write an explicit formula for ana_{n}an​, the nth n^{\text {th }}nth term of the sequence 8,−32,128,…8,-32,128, \ldots8,−32,128,….

Problem 153

Chapter 1∗1^{*}1∗ \qquad 28 points / ALL WORK FOR CREDIT! Find the sum of the first 42 terms in the following sequence: 11,17,23,…11,17,23, \ldots11,17,23,…

Problem 154

Problem 155

∑n=1∞2n+2\sum_{n=1}^{\infty} \frac{2}{\sqrt{n}+2}∑n=1∞​n​+22​

Problem 156

Problem 157

(11) These numbers make up two linear sequences.1 3 4 5 7 7 10 13 What are the two linear sequences? \qquad \qquad \qquad 2nd 2^{\text {nd }}2nd \qquad \qquad \qquad \qquad

Find the common difference $d$ of each arithmetic sequence. 1) $\{5,8,11,14,17, \ldots\}$ $\square$ Check Show answer 2) $\{4,0,-4,-8,-12, \ldots\}$ $\square$ Check
Show answer

8 Mark for Review 40
The radius of convergence of the power series $\sum_{n=0}^{\infty} \frac{n!(n+1)!}{(2 n)!} x^{n}$ is (A) 0
B 1 (C) 2 (D) 4

Determine whether the given series is convergent or divergent. $\sum_{n=1}^{\infty} \frac{n}{\sqrt{2 n^{2}+1}}$ convergent divergent

8. Find the indicated arithmetic term. a) $a=5, d=3$ ; find $t_{12}$ c) $a=-\frac{3}{4}, d=\frac{1}{2}$ ; find $t_{10}$ e) $a=-0.75, d=0.05$ ; find $t_{40}$

Find the nth , or general, term for the geometric sequence. $2,-2,2,-2, \ldots$
Write an expression for the nth term.

Ex: $\sum_{n=1}^{\infty} \frac{(x-3)^{n}}{5^{n}(n+1)}$

Agebraic Thinking: Look at the patiern below. $8,21,34,47,60$
What could be a rule for this patfern? A) Add 13 B) Add 15 C) Add II - $)$ Subtract 13

Question Watch
Find the common ratio of the geometric sequence $20,160,1280, \ldots$
Answer Attempt 2 out of 2 $\square$ Submit Answer

Write the arithmetic sequence $-17,-13,-9,-5, \ldots$ in the standard form: $a_{n}=$ $\square$

Write the arithmetic sequence $-3,-10,-17,-24, \ldots$ in the standard form: $a_{n}=$ $\square$

check the Cenvergence of the series. then calculate the radius of Convergence. Find the radius of convergence using derivatives and integration by parts. (1) $\sum_{n=0}^{\infty} \frac{z^{n}}{n!}$ (2) $\sum_{n=0}^{\infty} z^{n}$

Calculate the series: $\sum_{n=1}^{\infty} n\left(\frac{i}{1-i}\right)^{n-1}$

A section in a stadium has 24 seats in the first row, 28 seats in the second row, increasing by 4 seats each row for a total of 32 rows. How many seats are in this section of the stadium? $\square$ seats

write the taylor expansion of function: $f(z)=\frac{1}{3+z}$ around the Point: $z_{0}=2 i$

4. If the sum is 220 and the first term is 10 , find the common difference if the last term is 30. A. 2 B. 5 C. 3 D. $2 / 3$

5. Determine the sum of the progression if there are 7 arithmetic mean between 3 and 35.
171 B. 182 C. 232 D. 216

Write an explicit formula for $a_{n}$ , the $n^{\text {th }}$ term of the sequence $15,7,-1, \ldots$

Write an explicit formula for $a_{n}$ , the $n^{\text {th }}$ term of the sequence $8,-32,128, \ldots$ .

Chapter $1^{*}$ $\qquad$ 28 points / ALL WORK FOR CREDIT! Find the sum of the first 42 terms in the following sequence: $11,17,23, \ldots$

$\sum_{n=1}^{\infty} \frac{2}{\sqrt{n}+2}$

(11) These numbers make up two linear sequences.
1 3 4 5 7 7 10 13 What are the two linear sequences? $\qquad$ $\qquad$ $\qquad$ $2^{\text {nd }}$ $\qquad$ $\qquad$ $\qquad$ $\qquad$

reach series will converge or diverge. Then sible. 7) $8.1+3.24+1.296+0.5184 \ldots$

Determine whether the series $\sum_{k=0}^{\infty} \frac{k}{\sqrt{k^{2}+3}}$ converges.

Evaluate the infinite geometric series: $1+1 / 5+1 / 25+\ldots$
1 2 3
65/27 5/4 $9 / 5$ 5/6

What are the first three terms of the arithmetic sequence $a_{n}=-4-7(n-1)$ ? $\begin{array}{l} a_{1}=\text { Ex:10 } 0 \\ a_{2}=\square \\ a_{3}=\square \end{array}$

27-28 Explain why the Integral Test can't be used to determine whether the series is convergent.
27. $\sum_{n=1}^{\infty} \frac{\cos \pi n}{\sqrt{n}}$
28. $\sum_{n=1}^{\infty} \frac{\cos ^{2} n}{1+n^{2}}$

Question Watch Video Show Examples
Write an explicit formula for $a_{n}$ , the $n^{\text {th }}$ term of the sequence $2,-10,50, \ldots$
Answer Attempt 2 out of 2 $a_{n}=$ $\square$ Submit Answer Still Stuck?

57-63 Calcule los valores de $x$ para los cuales la serie converge. Determine la suma de la serie para dichos valores de $x$ .
57. $\sum_{n=1}^{\infty}(-5)^{n} x^{n}$
58. $\sum_{n=1}^{\infty}(x+2)^{n}$

2-30 Determine si la serie es absolutamente convergente, condicionalmente convergente o divergente.
3. $\sum_{n=1}^{\infty} \frac{n}{5^{n}}$
4. $\sum_{n=1}^{\infty}(-1)^{n-1} \frac{n}{n^{2}+4}$

Find the eighth term of the geometric sequence, given the first term and common ratio. $\begin{array}{c} a_{1}=6 \text { and } r=-1 / 3 \\ -\frac{[?]}{[\quad} \end{array}$

Find the fourth term of the geometric sequence, given the first term and common ratio. $a_{1}=8 \text { and } r=4$ $\square$