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23.3: Exercises

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    Exercise \(\PageIndex{1}\)

    Find the first seven terms of the sequence.

    1. \(a_n=3n\)
    2. \(a_n=5n+3\)
    3. \(a_n=n^2+2\)
    4. \(a_n=n\)
    5. \(a_n=(-1)^{n+1}\)
    6. \(a_n=\dfrac{\sqrt{n+1}}{n}\)
    7. \(a_k=10^k\)
    8. \(a_i=5+(-1)^i\)
    Answer
    1. \(3, 6, 9, 12, 15, 18, 21\)
    2. \(8, 13, 18, 23, 28, 33, 38\)
    3. \(3, 6, 11, 18, 27, 38, 51\)
    4. \(1, 2, 3, 4, 5, 6, 7\)
    5. \(1, −1, 1, −1, 1, −1, 1\)
    6. \(\sqrt{2}, \dfrac{\sqrt{3}}{2}, \dfrac{2}{3}, \dfrac{\sqrt{5}}{4}, \dfrac{\sqrt{6}}{5}, \dfrac{\sqrt{7}}{6}, \dfrac{\sqrt{8}}{7}\)
    7. \(10, 100, 1000, 10000, 100000, 1000000, 10000000\)
    8. \(4, 6, 4, 6, 4, 6, 4\)

    Exercise \(\PageIndex{2}\)

    Find the first six terms of the sequence.

    1. \(a_1=5\), \(a_n=a_{n-1}+3\) for \(n\geq 2\)
    2. \(a_1=7\), \(a_n=10\cdot a_{n-1}\) for \(n\geq 2\)
    3. \(a_1=1\), \(a_n=2\cdot a_{n-1}+1\) for \(n\geq 2\)
    4. \(a_1=6\), \(a_2=4\), \(a_n=a_{n-1}-a_{n-2}\) for \(n\geq 3\)
    Answer
    1. \(5, 8, 11, 14, 17\)
    2. \(7, 70, 700, 7000, 70000\)
    3. \(1, 3, 7, 15, 31\)
    4. \(6, 4, −2, −6, −4\)

    Exercise \(\PageIndex{3}\)

    Find the value of the series.

    1. \(\sum_{n=1}^4 a_n\), where \(a_n=5n\)
    2. \(\sum_{k=1}^5 a_k\), where \(a_k=k\)
    3. \(\sum_{i=1}^4 a_i\), where \(a_n=n^2\)
    4. \(\sum_{n=1}^6 (n-4)\)
    5. \(\sum_{k=1}^3 (k^2+4k-4)\)
    6. \(\sum_{j=1}^4 \dfrac{1}{j+1}\)
    Answer
    1. \(50\)
    2. \(15\)
    3. \(30\)
    4. \(−3\)
    5. \(26\)
    6. \(\dfrac {77}{60}\)

    Exercise \(\PageIndex{4}\)

    Is the sequence below part of an arithmetic sequence? In the case that it is part of an arithmetic sequence, find the formula for the \(n\)th term \(a_n\) in the form \(a_n=a_1+d\cdot (n-1)\).

    1. \(5, 8, 11, 14, 17, \dots\)
    2. \(-10, -7, -4, -1, 2, \dots\)
    3. \(-1, 1, -1, 1, -1, 1, \dots\)
    4. \(18, 164, 310, 474, \dots\)
    5. \(73.4, 51.7, 30, \dots\)
    6. \(9, 3, -3, -8, -14, \dots\)
    7. \(4, 4, 4, 4, 4, \dots\)
    8. \(-2.72, -2.82, -2.92, -3.02, -3.12, \dots\)
    9. \(\sqrt{2}, \sqrt{5}, \sqrt{8}, \sqrt{11}, \dots\)
    10. \(\dfrac{-3}{5}, \dfrac{-1}{10}, \dfrac{2}{5}, \dots\)
    11. \(a_n=4+5\cdot n\)
    12. \(a_j=2\cdot j-5\)
    13. \(a_n=n^2 +8n+15\)
    14. \(a_k=9\cdot (k+5) +7k-1\)
    Answer

    For the convenience of those who prefer to use \(a_{n}=a+b \cdot n\) as standard form we have provided answers also in that form.

    1. \(5 + 3(n−1) = 2 + 3n\)
    2. \(−10 + 3(n−1) = −13 + 3n\)
    3. no
    4. no
    5. \(73.4−21.7(n−1) = 95.1−21.7n\)
    6. no
    7. \(4 + 0 ·(n−1) = 4 + 0 ·n\)
    8. \(-2.72-.1(n-1)=-2.62-.1 n\)
    9. no
    10. \(-\dfrac{3}{5}+\dfrac{1}{2}(n-1)=-\dfrac{11}{10}+\dfrac{1}{2} n\)
    11. \(9 + 5(n − 1) = 4 + 5n\)
    12. \(−3 + 2(j − 1) = −5 + 2j\)
    13. no
    14. \(29 + 16(k − 1) = 13 + 16k\)

    Exercise \(\PageIndex{5}\)

    Determine the general \(n\)th term \(a_n\) of an arithmetic sequence \(\{a_n\}\) with the data given below.

    1. \(d=4\), and \(a_{8}=57\)
    2. \(d=-3\), and \(a_{99}=-70\)
    3. \(a_1=14\), and \(a_{7}=-16\)
    4. \(a_1=-80\), and \(a_{5}=224\)
    5. \(a_{3}=10\), and \(a_{14}=-23\)
    6. \(a_{20}=2\), and \(a_{60}=32\)
    Answer
    1. \(57 + 4(n − 8) = 29 + 4(n − 1) = 25 + 4n\)
    2. \(−70 − 3(n − 99) = 224 − 3(n − 1) = 227 − 3n\)
    3. \(14 − 5(n − 1) = 19 − 5n\)
    4. \(−80 + 76(n − 1) = −156 + 76n\)
    5. \(10 − 3(n − 3) = 16 − 3(n − 1) = 19 − 3n\)
    6. \(2+\dfrac{3}{4}(n-20)=-49 / 4+\dfrac{3}{4}(n-1)=-13+\dfrac{3}{4} n\)

    Exercise \(\PageIndex{6}\)

    Determine the value of the indicated term of the given arithmetic sequence.

    1. if \(a_1=8\), and \(a_{15}=92\), find \(a_{19}\)
    2. if \(d=-2\), and \(a_3=31\), find \(a_{81}\)
    3. if \(a_1=0\), and \(a_{17}=-102\), find \(a_{73}\)
    4. if \(a_{7}=128\), and \(a_{37}=38\), find \(a_{26}\)
    Answer
    1. \(116\)
    2. \(187\)
    3. \(-\dfrac{3621}{8}\)
    4. \(71\)

    Exercise \(\PageIndex{7}\)

    Determine the sum of the arithmetic sequence.

    1. Find the sum \(a_1+\dots +a_{48}\) for the arithmetic sequence \(a_i=4i+7\).
    2. Find the sum \(\sum_{i=1}^{21}a_i\) for the arithmetic sequence \(a_n=2-5n\).
    3. Find the sum: \(\sum\limits_{i=1}^{99} (10\cdot i+1)\)
    4. Find the sum: \(\sum\limits_{n=1}^{200} (-9-n)\)
    5. Find the sum of the first \(100\) terms of the arithmetic sequence: \(2, 4, 6, 8, 10, 12, \dots\)
    6. Find the sum of the first \(83\) terms of the arithmetic sequence: \(25, 21, 17, 13, 9, 5, \dots\)
    7. Find the sum of the first \(75\) terms of the arithmetic sequence: \(2012, 2002, 1992, 1982, \dots\)
    8. Find the sum of the first \(16\) terms of the arithmetic sequence: \(-11, -6, -1, \dots\)
    9. Find the sum of the first \(99\) terms of the arithmetic sequence: \(-8, -8.2, -8.4, -8.6, -8.8, -9, -9.2, \dots\)
    10. Find the sum \(7+8+9+10+\dots+776+777\)
    11. Find the sum of the first \(40\) terms of the arithmetic sequence: \(5, 5, 5, 5, 5, \dots\)
    Answer
    1. \(5, 040\)
    2. \(−1, 113\)
    3. \(49, 599\)
    4. \(−21, 900\)
    5. \(10, 100\)
    6. \(−11, 537\)
    7. \(123, 150\)
    8. \(424\)
    9. \(−1762.2\)
    10. \(302, 232\)
    11. \(200\)

    This page titled 23.3: 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.