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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

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) $29$ . $91$ $\newline$ (D) $31$ . $98$

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Sum of finite series not start from 1

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)

29

.

91

\newline

(D)

31

.

98

Identify Terms and Ratio: Identify the first term $a_1$ , common ratio $r$ , and number of terms $n$ in the geometric series. $\newline$ The first term $a_1$ is $8$ , the second term is $6$ , and the third term is $4.5$ . To find the common ratio $r$ , we divide the second term by the first term. $\newline$ $r = \frac{6}{8} = 0.75$ $\newline$ The number of terms $n$ is given as $r$ $0$ .
Use Sum Formula: Use the formula for the sum of the first $n$ terms of a geometric series: $S_n = a_1 \times (1 - r^n) / (1 - r)$ , where $S_n$ is the sum of the first $n$ terms, $a_1$ is the first term, $r$ is the common ratio, and $n$ is the number of terms. $\newline$ Plug in the values we have: $a_1 = 8$ , $r = 0.75$ , and $n = 25$ . $\newline$ $S_n = a_1 \times (1 - r^n) / (1 - r)$ $0$
Calculate Sum: Calculate the sum using the values from the previous step. $\newline$ $S_{25} = 8 \times (1 - 0.75^{25}) / (1 - 0.75)$ $\newline$ $S_{25} = 8 \times (1 - 0.75^{25}) / 0.25$ $\newline$ Since $0.75^{25}$ is a very small number, we can approximate it to $0$ for the sake of simplicity in calculation. $\newline$ $S_{25} \approx 8 \times (1 - 0) / 0.25$ $\newline$ $S_{25} \approx 8 / 0.25$ $\newline$ $S_{25} \approx 32$
Check Answer Choices: Check the answer choices to see which one matches our calculated sum. The closest answer to our calculated sum of $32$ is (D) $31.98$ .

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