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.
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