Question
Write the geometric series with a common ratio of 3, an initial value of 10, and 100 terms in sigma notation.
Answers
Answered by
GPT-4o mini
To express the geometric series with a common ratio of 3, an initial value of 10, and 100 terms in sigma notation, we can use the formula for the \( n \)-th term of a geometric series:
\[
a_n = a \cdot r^{n-1}
\]
where:
- \( a \) is the initial value (10),
- \( r \) is the common ratio (3),
- \( n \) is the term number.
In this case, the \( n \)-th term of the series can be expressed as:
\[
a_n = 10 \cdot 3^{n-1}
\]
To write the series in sigma notation, we sum the first 100 terms. Thus, the sigma notation for this series is:
\[
\sum_{n=1}^{100} 10 \cdot 3^{n-1}
\]
This notation represents the sum of 100 terms of the geometric series starting from \( n=1 \) to \( n=100 \).
\[
a_n = a \cdot r^{n-1}
\]
where:
- \( a \) is the initial value (10),
- \( r \) is the common ratio (3),
- \( n \) is the term number.
In this case, the \( n \)-th term of the series can be expressed as:
\[
a_n = 10 \cdot 3^{n-1}
\]
To write the series in sigma notation, we sum the first 100 terms. Thus, the sigma notation for this series is:
\[
\sum_{n=1}^{100} 10 \cdot 3^{n-1}
\]
This notation represents the sum of 100 terms of the geometric series starting from \( n=1 \) to \( n=100 \).