13.4E: Series and Their Notations (Exercises)
21. Use summation notation to write the sum of terms \(\frac{1}{2} m+5\) from \(m=0\) to \(m=5\).
22. Use summation notation to write the sum that results from adding the number 13 twenty times.
23. Use the formula for the sum of the first \(n\) terms of an arithmetic series to find the sum of the first eleven terms of the arithmetic series \(2.5,4,5.5, \ldots\)
24. A ladder has 15 tapered rungs, the lengths of which increase by a common difference. The first rung is 5 inches long, and the last rung is 20 inches long. What is the sum of the lengths of the rungs?
25. Use the formula for the sum of the first \(n\) terms of a geometric series to find \(S_{9}\) for the series \(12,6,3, \frac{3}{2}, \ldots\)
26. The fees for the first three years of a hunting club membership are given in Table 1 . If fees continue to rise at the same rate, how much will the total cost be for the first ten years of membership?
| Year | Membership Fees |
|---|---|
| 1 | $1500 |
| 2 | $1950 |
| 3 | $2535 |
27. Find the sum of the infinite geometric series \(\sum_{k=1}^{\infty} 45 \cdot\left(-\frac{1}{3}\right)^{k-1}\).
28. A ball has a bounce-back ratio of \(\frac{3}{5}\) the height of the previous bounce. Write a series representing the total distance traveled by the ball, assuming it was initially dropped from a height of 5 feet. What is the total distance? (Hint: the total distance the ball travels on each bounce is the sum of the heights of the rise and the fall.).
29. Alejandro deposits \(\$ 80\) of his monthly earnings into an annuity that earns \(6.25 \%\) annual interest, compounded monthly. How much money will he have saved after 5 years?
30. The twins Sarah and Scott both opened retirement accounts on their \(21^{\text {st }}\) birthday. Sarah deposits \(\$ 4,800.00\) each year, earning \(5.5 \%\) annual interest, compounded monthly. Scott deposits \(\$ 3,600.00\) each year, earning \(8.5 \%\) annual interest, compounded monthly. Which twin will earn the most interest by the time they are 55 years old? How much more?