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

15,30,60,dots
Answer:

Find the sum of the first $10$ terms of the following series, to the nearest integer. $\newline$ $15,30,60, \ldots$ $\newline$ 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

15,30,60, \ldots

\newline

Answer:

Identify type of series: Identify the type of series. $\newline$ The given series is geometric because each term is multiplied by a common ratio to get the next term.
Determine common ratio: Determine the common ratio $r$ of the series. $\newline$ To find the common ratio, divide the second term by the first term. $\newline$ $\frac{30}{15} = 2$ $\newline$ Common Ratio $r$ : $2$
Use formula for sum: 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.
Plug values into formula: Plug the values into the formula to find the sum of the first $10$ terms. $a = 15$ (the first term) $r = 2$ (the common ratio) $n = 10$ (the number of terms) $S_{10} = 15 \times (1 - 2^{10}) / (1 - 2)$
Calculate sum: Calculate the sum using the values from Step $4$ . $\newline$ $S_{10} = 15 \times (1 - 1024) / (1 - 2)$ $\newline$ $S_{10} = 15 \times (-1023) / (-1)$ $\newline$ $S_{10} = 15 \times 1023$ $\newline$ $S_{10} = 15345$
Round sum: Round the sum to the nearest integer. The sum is already an integer, so no rounding is necessary.

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