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.

Identify terms and ratio: Identify the first term (a1) and the common ratio (r) of the geometric series. The first term (a1) 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 = a1 * (1 - r^n) / (1 - r), where n is the number of terms. We need to find the sum of the first 4 terms, so n = 4.Substitute values: Substitute the values of a1, r, and n into the formula to calculate the sum. S_4 = 64 * (1 - 0.75^4) / (1 - 0.75)Calculate 0.75^4: Calculate the value of 0.75^4. 0.75^4 = 0.31640625Substitute back into formula: Substitute the value of 0.75^4 back into the sum formula. S_4 = 64 * (1 - 0.31640625) / (1 - 0.75)Calculate numerator: Calculate the numerator of the sum formula. 1 - 0.31640625 = 0.68359375Calculate denominator: Calculate the denominator of the sum formula. 1 - 0.75 = 0.25Divide for sum: Divide the numerator by the denominator to find the sum of the first 4 terms. S_4 = 64 * 0.68359375 / 0.25Perform final calculation: Perform the final calculation. S_4 = 64 * 2.734375 S_4 = 175.0

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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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Math Problems

Algebra 2

Find the sum of a finite geometric series

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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. $\newline$ $S_4 = 64 \times 2.734375$ $\newline$ $S_4 = 175.0$