Question

find the partial sum, $s_8$, for the geometric sequence with $a = 2, r = 4$.
$s_8 = \square$
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Explanation:

Step1: Recall Geometric Series Sum Formula

The formula for the partial sum \( S_n \) of a geometric sequence is \( S_n = a \cdot \frac{r^n - 1}{r - 1} \) when \( r
eq 1 \), where \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms.
Here, \( a = 2 \), \( r = 4 \), and \( n = 8 \).

Step2: Substitute Values into Formula

Substitute \( a = 2 \), \( r = 4 \), and \( n = 8 \) into the formula:
\( S_8 = 2 \cdot \frac{4^8 - 1}{4 - 1} \)

Step3: Calculate \( 4^8 \)

First, calculate \( 4^8 \). \( 4^2 = 16 \), \( 4^4=(4^2)^2 = 16^2 = 256 \), \( 4^8=(4^4)^2 = 256^2 = 65536 \).

Step4: Simplify Numerator and Denominator

The numerator is \( 65536 - 1 = 65535 \), and the denominator is \( 4 - 1 = 3 \). Then, \( \frac{65535}{3}=21845 \).

Step5: Multiply by \( a \)

Multiply this result by \( a = 2 \): \( S_8 = 2 \times 21845 = 43690 \).

Answer:

\( 43690 \)