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Find the sum of the first 7 terms of the following series, to the nearest integer.

25,100,400,dots
Answer:

Find the sum of the first $7$ terms of the following series, to the nearest integer. $\newline$ $25,100,400, \ldots$ $\newline$ Answer:

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Geometric sequences

Full solution

Q. Find the sum of the first

7

terms of the following series, to the nearest integer.

\newline

25,100,400, \ldots

\newline

Answer:

Identify type and ratio: Identify the type of series and the common ratio. $\newline$ The given series is geometric because each term after the first is obtained by multiplying the previous term by a constant number. $\newline$ To find the common ratio $r$ , we divide the second term by the first term. $\newline$ $\frac{100}{25} = 4$ $\newline$ Common Ratio: $4$
Use sum formula: Use the formula for the sum of the first $n$ terms of a geometric series. $\newline$ The formula for the sum of the first $n$ terms ( $S_n$ ) of a geometric series is: $\newline$ $S_n = a \times (1 - r^n) / (1 - r)$ , where $a$ is the first term, $r$ is the common ratio, and $n$ is the number of terms. $\newline$ In this case, $a = 25$ , $r = 4$ , and $n = 7$ .
Substitute and calculate: Substitute the values into the formula and calculate the sum. $\newline$ $S_7 = 25 \times (1 - 4^7) / (1 - 4)$ $\newline$ $S_7 = 25 \times (1 - 16384) / (1 - 4)$ $\newline$ $S_7 = 25 \times (-16383) / (-3)$ $\newline$ $S_7 = 25 \times 5461$ $\newline$ $S_7 = 136525$
Round to nearest integer: Round the sum to the nearest integer. $\newline$ The sum of the first $7$ terms is $136525$ , which is already an integer, so no rounding is necessary.

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