Sequences and Their Notations

For the following exercises, write the first four terms of the sequence.

7. [latex]{a}_{n}=-\frac{16}{n+1}[/latex]

11. [latex]{a}_{n}=1.25\cdot {\left(-4\right)}^{n - 1}[/latex]

13. [latex]{a}_{n}=\frac{{n}^{2}}{2n+1}[/latex]

For the following exercises, write an explicit formula for each sequence.

21. [latex]4, 7, 12, 19, 28,\dots[/latex]

23. [latex]1,1,\frac{4}{3},2,\frac{16}{5},\dots[/latex]

25. [latex]1,-\frac{1}{2},\frac{1}{4},-\frac{1}{8},\frac{1}{16},\dots[/latex]

For the following exercises, write the first five terms of the sequence.

27. [latex]{a}_{1}=3,\text{ }{a}_{n}=\left(-3\right){a}_{n - 1}[/latex]

29. [latex]{a}_{1}=-1,\text{ }{a}_{n}=\frac{{\left(-3\right)}^{n - 1}}{{a}_{n - 1}-2}[/latex]

For the following exercises, write a recursive formula for each sequence.

35. [latex]-8,-6,-3,1,6,\dots[/latex]

37. [latex]35,\text{ }38,\text{ }41,\text{ }44,\text{ }47,\text{ }\dots[/latex]

For the following exercises, write the first four terms of the sequence.

43. [latex]{a}_{n}=\frac{n!}{{n}^{\text{2}}}[/latex]

45. [latex]{a}_{n}=\frac{n!}{{n}^{2}-n - 1}[/latex]

For the following exercises, write a recursive formula for the sequence using the first five points shown on the graph.

55.

67. Consider the sequence defined by [latex]{a}_{n}=-6 - 8n[/latex]. Is [latex]{a}_{n}=-421[/latex] a term in the sequence? Verify the result.

Arithmetic Sequences

1. What is an arithmetic sequence?

3. How do we determine whether a sequence is arithmetic?

For the following exercises, determine whether the sequence is arithmetic. If so find the common difference.

7. [latex]\left\{0,\frac{1}{2},1,\frac{3}{2},2,...\right\}[/latex]

9. [latex]\left\{4,16,64,256,1024,...\right\}[/latex]

For the following exercises, write the first five terms of the arithmetic sequence.

11. [latex]{a}_{1}=0[/latex] , [latex]d=\frac{2}{3}[/latex]

13. [latex]{a}_{13}=-60,{a}_{33}=-160[/latex]

For the following exercises, find the specified term for the arithmetic sequence.

15. First term is 4, common difference is 5, find the 4^th term.

17. First term is 6, common difference is 7, find the 6^th term.

19. Find the first term or [latex]{a}_{1}[/latex] of an arithmetic sequence if [latex]{a}_{6}=12[/latex] and [latex]{a}_{14}=28[/latex].

21. Find the first term or [latex]{a}_{1}[/latex] of an arithmetic sequence if [latex]{a}_{8}=40[/latex] and [latex]{a}_{23}=115[/latex].

23. Find the first term or [latex]{a}_{1}[/latex] of an arithmetic sequence if [latex]{a}_{11}=11[/latex] and [latex]{a}_{21}=16[/latex].

For the following exercises, find the first five terms of the arithmetic sequence.

27. [latex]{a}_{1}=-19;\text{ }{a}_{n}={a}_{n - 1}-1.4[/latex]

41. [latex]{a}_{n}=24 - 4n[/latex]

For the following exercises, write a recursive formula for each arithmetic sequence.

29. [latex]{a}_{n}=\left\{17,26,35,...\right\}[/latex]

33. [latex]{a}_{n}=\left\{8.9,10.3,11.7,...\right\}[/latex]

35. [latex]{a}_{n}=\left\{\frac{1}{5},\frac{9}{20},\frac{7}{10},...\right\}[/latex]

37. [latex]{a}_{n}=\left\{\frac{1}{6},-\frac{11}{12},-2,...\right\}[/latex]

For the following exercises, write an explicit formula for each arithmetic sequence.

45. [latex]{a}_{n}=\left\{-5\text{, }95\text{, }195\text{, }...\right\}[/latex]

49. [latex]{a}_{n}=\left\{15.8,18.5,21.2,...\right\}[/latex]

51. [latex]{a}_{n}=\left\{0,\frac{1}{3},\frac{2}{3},...\right\}[/latex]

For the following exercises, find the number of terms in the given finite arithmetic sequence.

53. [latex]{a}_{n}=\left\{3\text{,}-4\text{,}-11\text{, }...\text{,}-60\right\}[/latex]

55. [latex]{a}_{n}=\left\{\frac{1}{2},2,\frac{7}{2},...,8\right\}[/latex]

Geometric Sequences

1. What is a geometric sequence?

3. What is the procedure for determining whether a sequence is geometric?

For the following exercises, determine whether the sequence is geometric. If so, find the common ratio.

7. [latex]-0.125,0.25,-0.5,1,-2,..[/latex].

9. [latex]-6,-12,-24,-48,-96,..[/latex].

11. [latex]-1,\frac{1}{2},-\frac{1}{4},\frac{1}{8},-\frac{1}{16},..[/latex].

13. [latex]0.8,4,20,100,500,..[/latex].

For the following exercises, write the first five terms of the geometric sequence.

15. [latex]\begin{array}{cc}{a}_{1}=5,& r=\frac{1}{5}\end{array}[/latex]

17. [latex]\begin{array}{cc}{a}_{6}=25,& {a}_{8}\end{array}=6.25[/latex]

23. [latex]\begin{array}{cc}{a}_{1}=7,& {a}_{n}=0.2{a}_{n - 1}\end{array}[/latex]

33. [latex]{a}_{n}=12\cdot {\left(-\frac{1}{2}\right)}^{n - 1}[/latex]

For the following exercises, find the specified term for the geometric sequence.

19. The first term is 16 and the common ratio is [latex]-\frac{1}{3}[/latex]. Find the 4^th term.

21. [latex]{a}_{n}=\left\{-2,\frac{2}{3},-\frac{2}{9},\frac{2}{27},...\right\}[/latex]. Find [latex]{a}_{7}[/latex].

43. Let [latex]{a}_{n}=-{\left(-\frac{1}{3}\right)}^{n - 1}[/latex]. Find [latex]{a}_{12}[/latex].

For the following exercises, write a recursive formula for each geometric sequence.

25. [latex]{a}_{n}=\left\{-32,-16,-8,-4,...\right\}[/latex]

27. [latex]{a}_{n}=\left\{10,-3,0.9,-0.27,...\right\}[/latex]

29. [latex]{a}_{n}=\left\{\frac{3}{5},\frac{1}{10},\frac{1}{60},\frac{1}{360},...\right\}[/latex]

31. [latex]{a}_{n}=\left\{\frac{1}{512},-\frac{1}{128},\frac{1}{32},-\frac{1}{8},...\right\}[/latex]

For the following exercises, write an explicit formula for each geometric sequence.

35. [latex]{a}_{n}=\left\{1,3,9,27,...\right\}[/latex]

37. [latex]{a}_{n}=\left\{0.8,-4,20,-100,...\right\}[/latex]

39. [latex]{a}_{n}=\left\{-1,-\frac{4}{5},-\frac{16}{25},-\frac{64}{125},...\right\}[/latex]

41. [latex]{a}_{n}=\left\{3,-1,\frac{1}{3},-\frac{1}{9},...\right\}[/latex]

For the following exercises, find the number of terms in the given finite geometric sequence.

45. [latex]{a}_{n}=\left\{2,1,\frac{1}{2},...,\frac{1}{1024}\right\}[/latex]

51. Use recursive formulas to give two examples of geometric sequences whose 3^rd terms are [latex]200[/latex].

53. Find the 5^th term of the geometric sequence [latex]\left\{b,4b,16b,...\right\}[/latex].

Series and Their Notations

1. What is an [latex]n\text{th}[/latex] partial sum?

3. What is a geometric series?

For the following exercises, write the series using summation notation.

7. The sum from of [latex]n=0[/latex] to [latex]n=4[/latex] of [latex]5n[/latex]

9. The sum that results from adding the number 4 five times

11. [latex]10+18+26+\dots +162[/latex]

17. [latex]8+4+2+\dots +0.125[/latex]

For the following exercises, use the formula for the sum of the first [latex]n[/latex] terms of each arithmetic sequence.

13. [latex]\frac{3}{2}+2+\frac{5}{2}+3+\frac{7}{2}[/latex]

15. [latex]3.2+3.4+3.6+\dots +5.6[/latex]

35. [latex]6+\frac{15}{2}+9+\frac{21}{2}+12+\frac{27}{2}+15[/latex]

37. [latex]\sum _{k=1}^{11}\left(\frac{k}{2}-\frac{1}{2}\right)[/latex]

For the following exercises, use the formula for the sum of the first [latex]n[/latex] terms of each geometric sequence, and then state the indicated sum.

19. [latex]9+3+1+\frac{1}{3}+\frac{1}{9}[/latex]

21. [latex]\sum _{a=1}^{11}64\cdot {0.2}^{a - 1}[/latex]

39. [latex]{S}_{7}[/latex] for the series [latex]0.4 - 2+10 - 50..[/latex].

41. [latex]\sum _{n=1}^{10}-2\cdot {\left(\frac{1}{2}\right)}^{n - 1}[/latex]

For the following exercises, determine whether the infinite series has a sum. If so, write the formula for the sum. If not, state the reason.

23. [latex]2+1.6+1.28+1.024+..[/latex].

25. [latex]\sum_{k=1}^{\infty}-{\left(-\frac{1}{2}\right)}^{k - 1}[/latex]

For the following exercises, find the indicated sum.

31. [latex]\sum _{n=1}^{6}n\left(n - 2\right)[/latex]

33. [latex]\sum _{k=1}^{7}{2}^{k}[/latex]

For the following exercises, find the sum of the infinite geometric series.

43. [latex]-1-\frac{1}{4}-\frac{1}{16}-\frac{1}{64}..[/latex].

45. [latex]\sum _{n=1}^{\infty }4.6\cdot {0.5}^{n - 1}[/latex]

For the following exercises, determine the value of the annuity for the indicated monthly deposit amount, the number of deposits, and the interest rate.

47. Deposit amount: $150; total deposits: [latex]24[/latex]; interest rate: [latex]3%[/latex], compounded monthly

49. Deposit amount: $100; total deposits: [latex]120[/latex]; interest rate: [latex]10%[/latex], compounded semi-annually

57. Karl has two years to save [latex]$10,000[/latex] to buy a used car when he graduates. To the nearest dollar, what would his monthly deposits need to be if he invests in an account offering a 4.2% annual interest rate that compounds monthly?

59. A boulder rolled down a mountain, traveling 6 feet in the first second. Each successive second, its distance increased by 8 feet. How far did the boulder travel after 10 seconds?

61. A pendulum travels a distance of 3 feet on its first swing. On each successive swing, it travels [latex]\frac{3}{4}[/latex] the distance of the previous swing. What is the total distance traveled by the pendulum when it stops swinging?