Question

Find the sum of the infinite geometric sequence
459, 321.3, 224.91, 157.437, 110.2059
...

Sum =

Answer 1

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

The first term \( a \) is the first term of the sequence:

\[
a = 459
\]

Next, we need to find the common ratio \( r \). This is done by dividing the second term by the first term:

\[
r = \frac{321.3}{459}
\]

Calculating this:

\[
r = \frac{321.3}{459} \approx 0.699
\]

We can also verify this ratio by checking the ratio of subsequent terms:

\[
r = \frac{224.91}{321.3} \approx 0.699
\]
\[
r = \frac{157.437}{224.91} \approx 0.699
\]
\[
r = \frac{110.2059}{157.437} \approx 0.699
\]

Now we have the ratio \( r \approx 0.699 \), which is indeed consistent across several terms.

For an infinite geometric series to converge, the common ratio \( r \) must satisfy \( |r| < 1 \). Since \( 0.699 < 1 \), the series converges.

The formula for the sum \( S \) of an infinite geometric series is given by:

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

Substituting in our values:

\[
S = \frac{459}{1 - 0.699}
\]

Calculating the denominator:

\[
1 - 0.699 = 0.301
\]

Then, substituting back into the formula gives:

\[
S = \frac{459}{0.301} \approx 1525.916
\]

Thus, the sum of the infinite geometric sequence is:

\[
\boxed{1525.916}
\]