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The first term in a geometric series is 64 and the common ratio is 0.75 .
Find the sum of the first 4 terms in the series.

The first term in a geometric series is $64$ and the common ratio is $0$ . $75$ . $\newline$ Find the sum of the first $4$ terms in the series.

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Find the sum of a finite geometric series

Full solution

Q. The first term in a geometric series is

64

and the common ratio is

0

.

75

.

\newline

Find the sum of the first

4

terms in the series.

Identify terms and ratio: Identify the first term $a_1$ and the common ratio $r$ of the geometric series. $\newline$ The first term $a_1$ is given as $64$ , and the common ratio $r$ is given as $0.75$ .
Use sum formula: Use the formula for the sum of the first $n$ terms of a geometric series, which is $S_n = a_1 \times (1 - r^n) / (1 - r)$ , where $n$ is the number of terms. $\newline$ We need to find the sum of the first $4$ terms, so $n = 4$ .
Substitute values: Substitute the values of $a_1$ , $r$ , and $n$ into the formula to calculate the sum. $\newline$ $S_4 = 64 \times (1 - 0.75^4) / (1 - 0.75)$
Calculate $0.75^4$ : Calculate the value of $0.75^4$ . $\newline$ $0.75^4 = 0.31640625$
Substitute back into formula: Substitute the value of $0.75^4$ back into the sum formula. $\newline$ $S_4 = 64 \times (1 - 0.31640625) / (1 - 0.75)$
Calculate numerator: Calculate the numerator of the sum formula. $1 - 0.31640625 = 0.68359375$
Calculate denominator: Calculate the denominator of the sum formula. $\newline$ $1 - 0.75 = 0.25$
Divide for sum: Divide the numerator by the denominator to find the sum of the first $4$ terms. $S_4 = 64 \times 0.68359375 / 0.25$
Perform final calculation: Perform the final calculation. $\newline$ $S_4 = 64 \times 2.734375$ $\newline$ $S_4 = 175.0$

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