Find the \( 6^{\text {th }} \) partial sum of the geometric sequence \( a_{n}=4^{n} \).
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Answer

\( S_{6} = 5460 \)

Steps

Step 1 ：\( S_{6} = \frac{a_{1}(1 - r^{6})}{1 - r} \)

Step 2 ：\( S_{6} = \frac{4^{1}(1 - 4^{6})}{1 - 4} \)

Step 3 ：\( S_{6} =\frac{4(1 - 4^{6})}{-3} \)

Step 4 ：\( S_{6} = -\frac{4(1 - 4^{6})}{3} = -\frac{4(1 - 4096)}{3} \)

Step 5 ：\( S_{6} = -\frac{4(-4095)}{3} = -\frac{-16380}{3} \)

Step 6 ：\( S_{6} = 5460 \)