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7.2E: Exercises

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    Exercises

    In Exercises 1 - 8, find the value of each sum using the definition of Summation Notation.

    1. \(\displaystyle \sum_{g = 4}^{9} (5g + 3)\)
    2. \(\displaystyle \sum_{k = 3}^{8} \frac{1}{k}\)
    3. \(\displaystyle \sum_{j = 0}^{5} 2^{j}\)
    4. \(\displaystyle \sum_{k = 0}^{2} (3k - 5)x^{k}\)
    5. \(\displaystyle \sum_{i = 1}^{4} \frac{1}{4}(i^{2} + 1)\)
    6. \(\displaystyle \sum_{n = 1}^{100} (-1)^{n}\)
    7. \(\displaystyle \sum_{n = 1}^{5} \frac{(n+1)!}{n!}\)
    8. \(\displaystyle \sum_{j = 1}^{3} \frac{5!}{j! \, (5-j)!}\)

    In Exercises 9 - 16, rewrite the sum using summation notation.

    1. \(8 + 11 + 14 + 17 + 20\)
    2. \(1 - 2 + 3 - 4 + 5 - 6 + 7 - 8\)
    3. \(x - \dfrac{x^{3}}{3} + \dfrac{x^{5}}{5} - \dfrac{x^{7}}{7}\)
    4. \(1 + 2 + 4 + \cdots + 2^{29} \vphantom{x - \dfrac{x^{3}}{3} + \dfrac{x^{5}}{5} - \dfrac{x^{7}}{7}}\)
    5. \(2 + \frac{3}{2} + \frac{4}{3} + \frac{5}{4} + \frac{6}{5}\)
    6. \(-\ln(3) + \ln(4) - \ln(5) + \cdots + \ln(20)\)
    7. \(1 - \frac{1}{4} + \frac{1}{9} - \frac{1}{16} + \frac{1}{25} - \frac{1}{36}\)
    8. \(\frac{1}{2}(x - 5) + \frac{1}{4}(x - 5)^{2} + \frac{1}{6}(x - 5)^{3} + \frac{1}{8}(x - 5)^{4}\)

    In Exercises 17 - 28, use the Properties of Summation Notation to find the sum.

    1. \(\displaystyle \sum_{n = 1}^{10} 5n+3\)
    2. \(\displaystyle \sum_{n = 1}^{20} 2n-1\)
    3. \(\displaystyle \sum_{k = 0}^{15} 3-k\)
    4. \(\displaystyle \sum_{n = 1}^{10} \left(\frac{1}{2}\right)^{n}\)
    5. \(\displaystyle \sum_{n = 1}^{5} \left(\frac{3}{2}\right)^{n}\)
    6. \(\displaystyle \sum_{k = 0}^{5} 2\left(\frac{1}{4}\right)^{k}\)
    7. \(1+4+7+ \ldots +295\)
    8. \(4+2+0-2- \ldots - 146\)
    9. \(1+3+9+ \ldots + 2187\)
    10. \(\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \ldots + \frac{1}{256}\vphantom{\displaystyle \sum_{n = 1}^{10} -2n + \left(\frac{5}{3}\right)^{n}}\)
    11. \(3 - \frac{3}{2} + \frac{3}{4} - \frac{3}{8}+- \dots +\frac{3}{256} \vphantom{\displaystyle \sum_{n = 1}^{10} -2n + \left(\frac{5}{3}\right)^{n}}\)
    12. \(\displaystyle \sum_{n = 1}^{10} -2n + \left(\frac{5}{3}\right)^{n}\)

    In Exercises 29 - 34, determine whether the infinite geometric series is convergent or divergent. If convergent, find the value the series converges to.

    1. \( 1 + \frac{1}{6} + \frac{1}{36} + \frac{1}{216} + \cdots \)
    2. \( \displaystyle \sum_{n = 1}^{\infty} \left( -\frac{4}{5} \right)^{n - 1}  \)
    3. \( \displaystyle \sum_{k = 0}^{\infty} \left( \frac{1}{3} \right)^{k}  \)
    4. \( \displaystyle \sum_{k = 0}^{\infty} \left( \frac{18}{7} \right)^{k}  \)
    5. \( \displaystyle \sum_{n = 1}^{\infty} \left( \frac{5}{8} \right)^{n + 3}  \)
    6. \( \displaystyle \sum_{n = 1}^{\infty} \left( \frac{3^{n - 1}}{8^n} \right)  \)

    In Exercises 35 - 38, use Theorem 7.2.4 to express each repeating decimal as a fraction of integers.

    1. \(0.\overline{7}\)
    2. \(0.\overline{13}\)
    3. \(10.\overline{159}\)
    4. \(-5.8\overline{67}\)

    In Exercises 39 - 45, use Theorem 7.2.3 to compute the future value of the annuity with the given terms. In all cases, assume the payment is made monthly, the interest rate given is the annual rate, and interest is compounded monthly.

    1. payments are $300, interest rate is 2.5%, term is 17 years.
    2. payments are $50, interest rate is 1.0%, term is 30 years.
    3. payments are $100, interest rate is 2.0%, term is 20 years
    4. payments are $100, interest rate is 2.0%, term is 25 years
    5. payments are $100, interest rate is 2.0%, term is 30 years
    6. payments are $100, interest rate is 2.0%, term is 35 years
    7. Suppose an ordinary annuity offers an annual interest rate of \(2 \%\), compounded monthly, for 30 years. What should the monthly payment be to have \(\$100,\!000\) at the end of the term?
    8. Prove the properties listed in Theorem 7.2.1.
    9. Show that the formula for the future value of an annuity due is \[A = P(1 + i)\left[\frac{(1 + i)^{nt} - 1}{i}\right]\nonumber\]
    10. Discuss with your classmates what goes wrong when trying to find the following sums.
      1. \(\displaystyle{ \sum_{k=1}^{\infty} 2^{k-1}}\)
      2. \(\displaystyle{ \sum_{k=1}^{\infty} (1.0001)^{k-1}}\)
      3. \(\displaystyle{ \sum_{k=1}^{\infty} (-1)^{k-1}}\)

    Answers

    1. \(213\)
    2. \(\frac{341}{280}\)
    3. \(63\)
    4. \(-5 - 2x + x^{2}\)
    5. \(\frac{17}{2}\)
    6. \(0\)
    7. \(20\)
    8. \(25\)
    9. \(\displaystyle \sum_{k = 1}^{5} (3k + 5)\)
    10. \(\displaystyle \sum_{k = 1}^{8} (-1)^{k - 1}k\)
    11. \(\displaystyle \sum_{k = 1}^{4} (-1)^{k - 1} \frac{x^{2k - 1}}{2k - 1}\)
    12. \(\displaystyle \sum_{k = 1}^{30} 2^{k-1}\)
    13. \(\displaystyle \sum_{k = 1}^{5} \frac{k + 1}{k}\)
    14. \(\displaystyle \sum_{k = 3}^{20} (-1)^{k} \ln(k)\)
    15. \(\displaystyle \sum_{k = 1}^{6} \frac{(-1)^{k - 1}}{k^{2}}\)
    16. \(\displaystyle \sum_{k = 1}^{4} \frac{1}{2k}(x - 5)^{k}\)
    17. \(305\)
    18. \(400\)
    19. \(-72\)
    20. \(\dfrac{1023}{1024}\)
    21. \(\dfrac{633}{32}\)
    22. \(\dfrac{1365}{512}\)
    23. \(14652\)
    24. \(-5396\)
    25. \(3280\)
    26. \(\dfrac{255}{256}\)
    27. \(\dfrac{513}{256}\)
    28. \(\dfrac{17771050}{59049}\)
    29.  
    30.  
    31.  
    32.  
    33.  
    34.  
    35. \(\dfrac{7}{9}\)
    36. \(\dfrac{13}{99}\)
    37. \(\dfrac{3383}{333}\)
    38. \(-\dfrac{5809}{990}\)
    39. $76,163.67
    40. \(\$20,\!981.40\)
    41. \(\$29,\!479.69\)
    42. \(\$38,\!882.12\)
    43. \(49,\!272.55\)
    44. \(60,\!754.80\)
    45. For \(\$100,\!000\), the monthly payment is \(\approx \$202.95\).

    7.2E: Exercises is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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