Find the sum of the first 25 terms in this geometric series: 8+6+4.5 dots Choose 1 answer: (A) 0.03 (B) 4.57 (c) 29.91 (D) 31.98

Identify first term: Identify the first term (a_1) of the geometric series. The first term is given as 8.Calculate common ratio: Identify the second term (a_2) and calculate the common ratio (r). The second term is given as 6. To find the common ratio, divide the second term by the first term. r = a_2 / a_1 = 6 / 8 = 0.75Use formula for sum: Use the formula for the sum of the first n terms of a geometric series. The formula is: S_n = a_1(1 - r^n) / (1 - r) We have: a_1 = 8 r = 0.75 n = 25Substitute values and calculate: Substitute the values into the formula and calculate the sum. S_25 = 8(1 - 0.75^25) / (1 - 0.75) Calculate 0.75^25 using a calculator. 0.75^25 ≈ 0.000316Continue calculation: Continue the calculation. S_25 = 8(1 - 0.000316) / (1 - 0.75) S_25 = 8(0.999684) / 0.25Complete calculation: Complete the calculation. S_25 = 8 * 0.999684 / 0.25 S_25 = 7.997472 / 0.25 S_25 = 31.989888Round the result: Round the result to two decimal places, as the answer choices are given in two decimal places. S_25 ≈ 31.99

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Math Problems

Algebra 2

Find the sum of a finite geometric series

Full solution

Q. Find the sum of the first

25

terms in this geometric series:

\newline

8+6+4.5 \ldots

\newline

Choose

1

answer:

\newline

(A)

0

03

\newline

(B)

4

57

\newline

(C)

\mathbf{2 9 . 9 1}

\newline

(D)

31

98

Identify first term: Identify the first term ( $a_1$ ) of the geometric series. $\newline$ The first term is given as $8$ .
Calculate common ratio: Identify the second term $a_2$ and calculate the common ratio $r$ . The second term is given as $6$ . To find the common ratio, divide the second term by the first term. $r = \frac{a_2}{a_1} = \frac{6}{8} = 0.75$
Use formula for sum: Use the formula for the sum of the first $n$ terms of a geometric series. $\newline$ The formula is: $\newline$ $S_n = \frac{a_1(1 - r^n)}{1 - r}$ $\newline$ We have: $\newline$ $a_1 = 8$ $\newline$ $r = 0.75$ $\newline$ $n = 25$
Substitute values and calculate: Substitute the values into the formula and calculate the sum. $\newline$ $S_{25} = \frac{8(1 - 0.75^{25})}{(1 - 0.75)}$ $\newline$ Calculate $0.75^{25}$ using a calculator. $\newline$ $0.75^{25} \approx 0.000316$
Continue calculation: Continue the calculation. $\newline$ $S_{25} = \frac{8(1 - 0.000316)}{1 - 0.75}$ $\newline$ $S_{25} = \frac{8(0.999684)}{0.25}$
Complete calculation: Complete the calculation. $\newline$ $S_{25} = 8 \times 0.999684 / 0.25$ $\newline$ $S_{25} = 7.997472 / 0.25$ $\newline$ $S_{25} = 31.989888$
Round the result: Round the result to two decimal places, as the answer choices are given in two decimal places. $\newline$ $S_{25} \approx 31.99$