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Chapter 22: Problem 663

Find the sum of the first sixteen terms of the arithmetic series whose first term is \(1 / 4\) and common difference is \(1 / 2\).

Short Answer

Expert verified
The sum of the first sixteen terms of the given arithmetic series is \(64\).

Step by step solution

01

1. Identify the given values

: We are given: First term (a1) = 1/4 Common difference (d) = 1/2 Number of terms (n) = 16
02

2. Find the last term of the series

: Using the formula for the n-th term of an arithmetic series, we have: a_n = a1 + (n-1) * d a_n = (1/4) + (16-1) * (1/2) a_n = (1/4) + 15 * (1/2) a_n = (1/4) + 15/2 a_n = 31/4
03

3. Find the sum of the series

: Now that we have the last term of the series, we can use the formula for the sum of an arithmetic series: Sum = (n * (a1 + a_n)) / 2 Sum = (16 * (1/4 + 31/4)) / 2 Sum = (16 * 32/4) / 2 Sum = (16 * 8) / 2 Sum = 128 / 2 Sum = 64
04

4. Conclusion

: The sum of the first sixteen terms of the given arithmetic series is 64.

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