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Find the sum of the first 9 terms in the following geometric series.
Do not round your answer.

64+32+16+dots

Find the sum of the first $9$ terms in the following geometric series. $\newline$ Do not round your answer. $\newline$ $64+32+16+\ldots$

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Distance between two points

Full solution

Q. Find the sum of the first

9

terms in the following geometric series.

\newline

Do not round your answer.

\newline

64+32+16+\ldots

Identify Terms and Ratio: Identify the first term $a$ and the common ratio $r$ of the geometric series. $\newline$ The first term $a = 64$ . $\newline$ To find the common ratio $r$ , divide the second term by the first term: $r = \frac{32}{64} = 0.5$ .
Use Sum Formula: Use the formula for the sum of the first $n$ terms of a geometric series: $S_n = \frac{a(1 - r^n)}{(1 - r)}$ , where $S_n$ is the sum of the first $n$ terms, $a$ is the first term, $r$ is the common ratio, and $n$ is the number of terms. $\newline$ Here, $n = 9$ , $a = 64$ , and $r = 0.5$ .
Calculate Substitution: Substitute the values into the formula and calculate the sum. $\newline$ $S_9 = 64(1 - 0.5^9) / (1 - 0.5)$ $\newline$ Calculate $0.5^9$ .
Calculate $0.5^9$ : $0.5^9 = 0.5 \times 0.5 \times 0.5 \times 0.5 \times 0.5 \times 0.5 \times 0.5 \times 0.5 \times 0.5$ $\newline$ $0.5^9 = 0.001953125$
Perform Subtraction and Division: Substitute $0.5^9$ into the formula and calculate the sum. $\newline$ $S_9 = 64(1 - 0.001953125) / (1 - 0.5)$ $\newline$ $S_9 = 64(0.998046875) / 0.5$
Perform Subtraction and Division: Substitute $0.5^9$ into the formula and calculate the sum. $\newline$ $S_9 = 64(1 - 0.001953125) / (1 - 0.5)$ $\newline$ $S_9 = 64(0.998046875) / 0.5$ Perform the subtraction and division to find the sum. $\newline$ $S_9 = 64 \times 0.998046875 / 0.5$ $\newline$ $S_9 = 63.875 / 0.5$ $\newline$ $S_9 = 127.75$

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