Question
Find the sum \( S_{7} \) of the first 7 terms of the geometric sequence with \( a_{1}=4 \) and \( r=4 \). Use the formula \( S_{n}=\frac{a_{1}\left(1-r^{n}\right)}{1-r} \) \[ S_{7}=\square \]
Ask by Vaughn Cole. in the United States
Mar 08,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( S_{7} = 21844 \)
Solution
Let \( a_1 = 4 \), \( r = 4 \), and \( n = 7 \).
We use the formula:
\[
S_n = \frac{a_1\left(1 - r^n\right)}{1 - r}
\]
Substitute the given values:
\[
S_{7} = \frac{4\left(1 - 4^7\right)}{1 - 4}
\]
Calculate \( 4^7 \):
\[
4^7 = 16384
\]
Now, substitute back:
\[
S_{7} = \frac{4\left(1 - 16384\right)}{1 - 4} = \frac{4\left(-16383\right)}{-3}
\]
Compute the numerator:
\[
4\left(-16383\right) = -65532
\]
Thus, the sum is:
\[
S_{7} = \frac{-65532}{-3} = 21844
\]
\[
S_{7} = 21844
\]
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To find the sum \( S_{7} \) of the first 7 terms of the geometric sequence with \( a_{1}=4 \) and \( r=4 \), we use the formula: \[ S_{n} = \frac{a_{1}(1 - r^n)}{1 - r} \] Substituting \( a_{1} = 4 \), \( r = 4 \), and \( n = 7 \) into the formula gives: \[ S_{7} = \frac{4(1 - 4^7)}{1 - 4} \] First, let's calculate \( 4^7 \): \[ 4^7 = 16384 \] Now, substitute this value back into the equation for \( S_{7} \): \[ S_{7} = \frac{4(1 - 16384)}{1 - 4} = \frac{4(-16383)}{-3} \] Calculating the sum: \[ S_{7} = \frac{4 \cdot 16383}{3} = \frac{65532}{3} = 21844 \] Thus, \[ S_{7} = 21844 \]
