Full solution

Q. The first term in a geometric series is

5

and the common ratio is

2

\newline

Find the sum of the first

10

terms in the series.

Geometric series formula: The sum of the first $n$ terms of a geometric series is given by the formula: $\newline$ $S_n = \frac{a_1(1 - r^n)}{1 - r}$ $\newline$ where $a_1$ is the first term, $r$ is the common ratio, and $n$ is the number of terms. $\newline$ For this series, $a_1 = 5$ , $r = 2$ , and $n = 10$ .
Calculate sum using formula: Calculate the sum of the first $10$ terms using the formula. $\newline$ Substitute the values into the formula: $\newline$ $S_{10} = \frac{5(1 - 2^{10})}{1 - 2}$
Simplify expression: Simplify the expression by calculating $2^{10}$ . $\newline$ $2^{10} = 1024$ $\newline$ Now substitute this value into the formula: $\newline$ $S_{10} = \frac{5(1 - 1024)}{1 - 2}$
Substitute values into formula: Simplify the numerator by subtracting $1024$ from $1$ . $\newline$ $1 - 1024 = -1023$ $\newline$ Now the formula looks like this: $\newline$ $S_{10} = \frac{5 \cdot (-1023)}{1 - 2}$
Simplify numerator: Simplify the denominator by subtracting $2$ from $1$ . $\newline$ $1 - 2 = -1$ $\newline$ Now the formula looks like this: $\newline$ $S_{10} = \frac{5 \cdot (-1023)}{-1}$
Simplify denominator: Divide the numerator by the denominator to find the sum. $\newline$ $S_{10} = -5 \times -1023$ $\newline$ $S_{10} = 5115$