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Find the sum of the first 10 terms of the following series, to the nearest integer.

21,126,756,dots
Answer:

Find the sum of the first $10$ terms of the following series, to the nearest integer. $\newline$ $21,126,756, \ldots$ $\newline$ Answer:

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Geometric sequences

Full solution

Q. Find the sum of the first

10

terms of the following series, to the nearest integer.

\newline

21,126,756, \ldots

\newline

Answer:

Identify Series Type: Identify whether the given series is geometric or arithmetic. In the given series, each term is a multiple of the previous term, which suggests that the series is geometric.
Find Common Ratio: Identify the common ratio of the geometric series. $\newline$ To find the common ratio, divide the second term by the first term. $\newline$ Common Ratio $r$ = $\frac{126}{21}$ = $6$
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: $\newline$ $S_n = a \cdot (1 - r^n) / (1 - r)$ , where $a$ is the first term and $r$ is the common ratio.
Calculate Sum: Calculate the sum of the first $10$ terms using the formula. $\newline$ First term $(a) = 21$ $\newline$ Common Ratio $(r) = 6$ $\newline$ Number of terms $(n) = 10$ $\newline$ $S_{10} = 21 \times (1 - 6^{10}) / (1 - 6)$
Perform Calculations: Perform the calculations. $\newline$ $S_{10} = 21 \times (1 - 60466176) / (1 - 6)$ $\newline$ $S_{10} = 21 \times (-60466175) / (-5)$ $\newline$ $S_{10} = 21 \times 12093235$ $\newline$ $S_{10} = 253958535$
Round Sum: Round the sum to the nearest integer. $\newline$ The sum is already an integer, so no rounding is necessary.

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