Question

Write the geometric series with the common ratio 3, initial value 10, and 100 terms in sigma notation.(2 points)

Answer 1

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:

\[
\sum_{n=1}^{100} 10 \cdot 3^{n-1}
\]