The geometric series you described has an initial value of 10, a common ratio of 3, and consists of 100 terms. The formula for the \( n \)-th term of a geometric series is given by:
\[ a_n = a_1 \cdot r^{n-1} \]
Where:
- \( a_1 \) is the first term,
- \( r \) is the common ratio,
- \( n \) is the term number.
In your case:
- \( a_1 = 10 \)
- \( r = 3 \)
The \( n \)-th term can be expressed as:
\[ a_n = 10 \cdot 3^{n-1} \]
To express the sum of the first 100 terms of this geometric series in sigma notation, you can write:
\[ S = \sum_{n=1}^{100} 10 \cdot 3^{n-1} \]
So, the geometric series in sigma notation is:
\[ \sum_{n=1}^{100} 10 \cdot 3^{n-1} \]