Question

Calculate the sum of the given geometric series: ∑6n=120(12)n−1
(1 point)
The sum is
.

Answer 1

The given geometric series is represented as:

\[
\sum_{n=1}^{20} 12^{n-1}
\]

This expression represents a geometric series with:
- The first term \( a = 12^0 = 1 \)
- The common ratio \( r = 12 \)
- The number of terms \( n = 20 \)

The sum \( S_n \) of the first \( n \) terms of a geometric series can be calculated using the formula:

\[
S_n = a \frac{r^n - 1}{r - 1}
\]

Plugging in the values:

- \( a = 1 \)
- \( r = 12 \)
- \( n = 20 \)

We get:

\[
S_{20} = 1 \cdot \frac{12^{20} - 1}{12 - 1} = \frac{12^{20} - 1}{11}
\]

Now, calculating \( 12^{20} \):

While we typically wouldn't calculate \( 12^{20} \) completely due to its large size, we can approximate or leave it in the exponent form:

If the exact number is not required and the sum in the form of \( \frac{12^{20} - 1}{11} \) suffices, we can express the final answer as:

\[
\text{The sum is } \frac{12^{20} - 1}{11}
\]

If you did want the actual value, you'd compute \( 12^{20} \) using a calculator or programming tool, but since this was not specified, we maintain the expression form.