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 Terms and Ratio: Identify the first term (a1), common ratio (r), and number of terms (n) in the geometric series. The first term a1 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. r = 6 / 8 = 0.75 The number of terms n is given as 25.Use Sum Formula: Use the formula for the sum of the first n terms of a geometric series: S_n = a1 * (1 - r^n) / (1 - r), where S_n is the sum of the first n terms, a1 is the first term, r is the common ratio, and n is the number of terms. Plug in the values we have: a1 = 8, r = 0.75, and n = 25. S_25 = 8 * (1 - 0.75^25) / (1 - 0.75)Calculate Sum: Calculate the sum using the values from the previous step. S_25 = 8 * (1 - 0.75^25) / (1 - 0.75) S_25 = 8 * (1 - 0.75^25) / 0.25 Since 0.75^25 is a very small number, we can approximate it to 0 for the sake of simplicity in calculation. S_25 ≈ 8 * (1 - 0) / 0.25 S_25 ≈ 8 / 0.25 S_25 ≈ 32Check 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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Math Problems

Algebra 2

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) $31.98$ .