Skip to main content
Mathematics LibreTexts

24.3: Exercises

  1. Last updated
  2. Save as PDF
  • Page ID
    54482

    • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    Exercise \(\PageIndex{1}\)

    Which of these sequences is geometric, arithmetic, or neither or both. Write the sequence in the usual form \(a_n=a_1+d(n-1)\) if it is an arithmetic sequence and \(a_n=a_1\cdot r^{n-1}\) if it is a geometric sequence.

    1. \(7, 14, 28, 56, \dots\)
    2. \(3, -30, 300, -3000, 30000, \dots\)
    3. \(81, 27, 9, 3, 1, \dfrac{1}{3}, \dots\)
    4. \(-7, -5, -3, -1, 1, 3, 5, 7, \dots\)
    5. \(-6, 2, -\dfrac 2 3, \dfrac 2 9, -\dfrac 2 {27}, \dots\)
    6. \(-2, -2\cdot \dfrac 2 3, -2 \cdot \left(\dfrac 2 3\right)^2, -2 \cdot \left(\dfrac 2 3\right)^3, \dots\)
    7. \(\dfrac 1 2, \dfrac 1 4, \dfrac 1 8, \dfrac 1 {16}, \dots\)
    8. \(2, 2, 2, 2, 2, \dots\)
    9. \(5, 1, 5, 1, 5, 1, 5, 1, \dots\)
    10. \(-2, 2, -2, 2, -2, 2, \dots\)
    11. \(0, 5, 10, 15, 20, \dots\)
    12. \(5, \dfrac 5 3, \dfrac 5 {3^2}, \dfrac 5 {3^3}, \dfrac 5 {3^4}, \dots\)
    13. \(\dfrac 1 2, \dfrac 1 4, \dfrac 1 8, \dfrac 1 {16}, \dots\)
    14. \(\log(2), \log(4), \log(8), \log(16), \dots\)
    15. \(a_n=-4^n\)
    16. \(a_n=-4n\)
    17. \(a_n=2\cdot (-9)^n\)
    18. \(a_n=\left(\dfrac 1 3 \right)^n\)
    19. \(a_n=-\left(\dfrac 5 7 \right)^n\)
    20. \(a_n=\left(-\dfrac 5 7 \right)^n\)
    21. \(a_n=\dfrac 2 n\)
    22. \(a_n=3n+1\)
    Answer
    1. geometric, \(7 \cdot 2^{n-1}\)
    2. geometric, \(3 \cdot(-10)^{n-1}\)
    3. geometric, \(81\left(\dfrac{1}{3}\right)^{n-1}\)
    4. arithmetic, \(-7+2(n-1)=-9+2 n\)
    5. geometric, \(-6\left(-\dfrac{1}{3}\right)^{n-1}\)
    6. geometric, \(-2\left(\dfrac{2}{3}\right)^{n-1}\)
    7. geometric, \(\dfrac{1}{2}\left(\dfrac{1}{2}\right)^{n-1}\)
    8. both, \(2=2+0(n-1)=2(1)^{n-1}\)
    9. neither
    10. geometric, \(-2(-1)^{n-1}\)
    11. arithmetic, \(5(n − 1)\)
    12. geometric, \(5\left(\dfrac{1}{3}\right)^{n-1}\)
    13. geometric, \(\dfrac{1}{2}\left(\dfrac{1}{2}\right)^{n-1}\)
    14. neither
    15. geometric, \(-4(4)^{n-1}\)
    16. arithmetic, \(−4 − 4(n − 1) = −4n\)
    17. geometric, \(-18(-9)^{n-1}\)
    18. geometric, \(\dfrac{1}{3}\left(\dfrac{1}{3}\right)^{n-1}\)
    19. geometric, \(-\dfrac{5}{7}\left(\dfrac{5}{7}\right)^{n-1}\)
    20. geometric, \(-\dfrac{5}{7}\left(-\dfrac{5}{7}\right)^{n-1}\)
    21. neither
    22. arithmetic, \(4+3(n-1)=1+3 n\)

    Exercise \(\PageIndex{2}\)

    A geometric sequence, \(a_n=a_1\cdot r^{n-1}\), has the given properties. Find the term \(a_n\) of the sequence.

    1. \(a_1=3\), and \(r=5\), find \(a_4\)
    2. \(a_1=200\), and \(r=-\dfrac 1 2\), find \(a_6\)
    3. \(a_1=-7\), and \(r=2\), find \(a_n\) (for all \(n\))
    4. \(r=2\), and \(a_4=48\), find \(a_1\)
    5. \(r=100\), and \(a_{4}=900,000\), find \(a_n\) (for all \(n\))
    6. \(a_{1}=20\), \(a_{4}=2500\), find \(a_n\) (for all \(n\))
    7. \(a_1=\dfrac 1 8\), and \(a_6=\dfrac{3^5}{8^6}\), find \(a_n\) (for all \(n\))
    8. \(a_3=36\), and \(a_{6}=972\), find \(a_n\) (for all \(n\))
    9. \(a_{8}=4000\), \(a_{10}=40\), and \(r\) is negative, find \(a_n\) (for all \(n\))
    Answer
    1. \(375\)
    2. \(6.25\)
    3. \(-7 \cdot 2^{n-1}\)
    4. \(6\)
    5. \(\dfrac{9}{10}(100)^{n-1}\)
    6. \(20 \cdot(5)^{n-1}\)
    7. \(\dfrac{1}{8}\left(\dfrac{3}{8}\right)^{n-1}\)
    8. \(4 \cdot 3^{n-1}\)
    9. \(-40000000000\left(-\dfrac{1}{10}\right)^{n-1}\)

    Exercise \(\PageIndex{3}\)

    Find the value of the finite geometric series using formula [EQU:geometric-series]. Confirm the formula either by adding the the summands directly, or alternatively by using the calculator.

    1. Find the sum \(\sum\limits_{j=1}^4 a_j\) for the geometric sequence \(a_j=5\cdot 4^{j-1}\).
    2. Find the sum \(\sum\limits_{i=1}^7 a_i\) for the geometric sequence \(a_n=\left(\dfrac 1 2\right)^n\).
    3. Find: \(\sum\limits_{m=1}^{5} \left(-\dfrac{1}{5}\right)^m\)
    4. Find: \(\sum\limits_{k=1}^{6} 2.7\cdot 10^k\)
    5. Find the sum of the first \(5\) terms of the geometric sequence: \[2, 6, 18, 54, \dots \nonumber \]
    6. Find the sum of the first \(6\) terms of the geometric sequence: \[-5, 15, -45, 135, \dots \nonumber \]
    7. Find the sum of the first \(8\) terms of the geometric sequence: \[-1, -7, -49, -343, \dots \nonumber \]
    8. Find the sum of the first \(10\) terms of the geometric sequence: \[600, -300, 150, -75, 37.5, \dots \nonumber \]
    9. Find the sum of the first \(40\) terms of the geometric sequence: \[5, 5, 5, 5, 5, \dots \nonumber \]
    Answer
    1. \(425\)
    2. \(\dfrac{127}{128}\)
    3. \(-\dfrac{521}{3125}\)
    4. \(2999997\)
    5. \(242\)
    6. \(910\)
    7. \(-960,800\)
    8. \(\dfrac{25,575}{64}\)
    9. \(200\)

    Exercise \(\PageIndex{4}\)

    Find the value of the infinite geometric series.

    1. \(\sum_{j=1}^\infty a_j\), for \(a_j=3\cdot \left(\dfrac 2 3\right)^{j-1}\)
    2. \(\sum_{j=1}^\infty 7\cdot \left(-\dfrac 1 5\right)^{j}\)
    3. \(\sum_{j=1}^\infty 6\cdot \dfrac 1 {3^j}\)
    4. \(\sum_{n=1}^\infty -2\cdot \left(0.8\right)^n\)
    5. \(\sum_{n=1}^\infty \left(0.99\right)^n\)
    6. \(27+9+3+1+\dfrac 1 3+\dots\)
    7. \(-2+1-\dfrac 1 2+\dfrac 1 4-\dots\)
    8. \(-6-2-\dfrac 2 3-\dfrac 2 9-\dots\)
    9. \(100+40+16+6.4+ \dots\)
    10. \(-54+18-6+2- \dots\)
    Answer
    1. \(9\)
    2. \(-\dfrac{7}{6}\)
    3. \(3\)
    4. \(-8\)
    5. \(99\)
    6. \(\dfrac{81}{2}\)
    7. \(-\dfrac{4}{3}\)
    8. \(-9\)
    9. \(\dfrac{500}{3}\)
    10. \(-\dfrac{81}{2}\)

    Exercise \(\PageIndex{5}\)

    Rewrite the decimal using an infinite geometric sequence, and then use the formula for infinite geometric series to rewrite the decimal as a fraction (see example 24.2.3).

    1. \(0.44444\dots\)
    2. \(0.77777\dots\)
    3. \(5.55555\dots\)
    4. \(0.2323232323\dots\)
    5. \(39.393939\dots\)
    6. \(0.248248248\dots\)
    7. \(20.02002\dots\)
    8. \(0.5040504\dots\)
    Answer
    1. \(\dfrac{4}{9}\)
    2. \(\dfrac{7}{9}\)
    3. \(\dfrac{50}{9}\)
    4. \(\dfrac{23}{99}\)
    5. \(\dfrac{1300}{33}\)
    6. \(\dfrac{248}{999}\)
    7. \(\dfrac{20000}{999}\)
    8. \(\dfrac{560}{1111}\)

    This page titled 24.3: Exercises is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Thomas Tradler and Holly Carley (New York City College of Technology at CUNY Academic Works) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.