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The common ratio in a geometric series is 0.5 and the first term is 256.
Find the sum of the first 6 terms in the series.

The common ratio in a geometric series is $0$ . $5$ and the first term is $256$ . $\newline$ Find the sum of the first $6$ terms in the series.

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Find the sum of a finite geometric series

Full solution

Q. The common ratio in a geometric series is

0

.

5

and the first term is

256

.

\newline

Find the sum of the first

6

terms in the series.

Question Prompt: The question prompt is: ＂Find the sum of the first $6$ terms in a geometric series with a common ratio of $0.5$ and a first term of $256$ .＂
Formula for Sum: To find the sum of the first $6$ terms of a geometric series, we use the formula for the sum of the first n terms of a geometric series, which is $S_n = \frac{a(1 - r^n)}{1 - r}$ , 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.
Substitute Values: Substitute the given values into the formula: $a = 256$ , $r = 0.5$ , and $n = 6$ . So, $S_6 = \frac{256(1 - (0.5)^6)}{1 - 0.5}$ .
Calculate Power: Calculate the value of $(0.5)^6$ . This equals $0.5^6 = 0.015625$ .
Substitute Back: Substitute the value of $(0.5)^6$ back into the formula: $S_6 = \frac{256(1 - 0.015625)}{1 - 0.5}$ .
Simplify Expression: Simplify the expression inside the parentheses and the denominator: $S_6 = \frac{256(0.984375)}{0.5}$ .
Perform Operations: Perform the multiplication and division to find $S_6$ : $S_6 = \frac{256 \times 0.984375}{0.5} = 512$ .

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