Question

Find the sum of the infinite geometric sequence
150,849, 150.849, 0.150849, 1.50849E-4, 1.50849E-7
...

Sum =

Answer 1

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

The first term, \( a \), is:
\[
a = 150849
\]

Next, we determine the common ratio \( r \) between the terms. We can do this by dividing the second term by the first term:
\[
r = \frac{150.849}{150849} \approx \frac{150849}{150849} \times 10^{-3} = 10^{-3} \text{ or } 0.001
\]

To check the common ratio further, we can divide the third term by the second term:
\[
r = \frac{0.150849}{150.849} \approx \frac{150849}{150849} \times 10^{-3} = 10^{-3} \text{ or } 0.001
\]

The same ratio applies when checking from the third to the fourth and fourth to the fifth terms, confirming that:
\[
r = 0.001
\]

Now we apply the formula for the sum \( S \) of an infinite geometric sequence:
\[
S = \frac{a}{1 - r}
\]
where \( |r| < 1 \).

Substituting the values we identified into the formula:
\[
S = \frac{150849}{1 - 0.001} = \frac{150849}{0.999}
\]
Calculating this yields:
\[
S \approx \frac{150849}{0.999} \approx 151000.849
\]

Therefore, the sum of the infinite geometric sequence is:
\[
\boxed{151000.849}
\]