Question

Find the sum of the infinite geometric sequence
304, 182.4, 109.44, 65.664, 39.3984
...

Sum =

Answer 1

To find the sum of the infinite geometric sequence, we need to identify the first term and the common ratio.

The first term \( a \) is:

\[
a = 304
\]

Next, we find the common ratio \( r \) by dividing the second term by the first term:

\[
r = \frac{182.4}{304} = 0.6
\]

Now we can verify that this ratio holds for the subsequent terms:

- \( \frac{109.44}{182.4} = 0.6 \)
- \( \frac{65.664}{109.44} = 0.6 \)
- \( \frac{39.3984}{65.664} = 0.6 \)

Since the common ratio \( r = 0.6 \) remains consistent across the terms, we confirm that it is indeed a geometric sequence.

The formula for the sum of an infinite geometric series, where \( |r| < 1 \), is given by:

\[
S = \frac{a}{1 - r}
\]

Substituting in our values:

\[
S = \frac{304}{1 - 0.6} = \frac{304}{0.4} = 760
\]

Thus, the sum of the infinite geometric sequence is:

\[
\boxed{760}
\]