Question
Write the sigma notation for the geometric series from the following finite geometric sequence: 3, 9, 27, 81, 243.(1 point) Responses ∑3n=15(3)n−1 sigma-summation underscript n equals 1 overscript 3 End Scripts 5 left parenthesis 3 right parenthesis superscript n minus 1 baseline ∑5n=15(3)n−1 sigma-summation underscript n equals 1 overscript 5 End Scripts 5 left parenthesis 3 right parenthesis superscript n minus 1 baseline ∑3n=13(3)n−1 sigma-summation underscript n equals 1 overscript 3 End Scripts 3 left parenthesis 3 right parenthesis superscript n minus 1 baseline ∑5n=13(3)n−1
Answers
GPT-4o mini
To find the sigma notation for the given finite geometric sequence: 3, 9, 27, 81, 243, let's first identify the parameters of the sequence.
1. The first term \( a = 3 \).
2. The common ratio \( r = \frac{9}{3} = 3 \).
3. The number of terms \( n = 5 \).
The general form for the \( n \)-th term of a geometric series can be expressed as:
\[
a_n = a \cdot r^{n-1}
\]
For this series, that translates to:
\[
a_n = 3 \cdot 3^{n-1} = 3^n
\]
We can represent the sum of this series in sigma notation as follows:
\[
\sum_{n=1}^{5} 3 \cdot 3^{n-1}
\]
Therefore, the correct option for the sigma notation of the finite geometric series from the given sequence is:
\[
\sum_{n=1}^{5} 3 \cdot 3^{n-1}
\]
Among the responses you provided, the correct answer is:
\(\sum_{n=1}^{5} 3(3)^{n-1}\) (which matches the calculated expression).
1. The first term \( a = 3 \).
2. The common ratio \( r = \frac{9}{3} = 3 \).
3. The number of terms \( n = 5 \).
The general form for the \( n \)-th term of a geometric series can be expressed as:
\[
a_n = a \cdot r^{n-1}
\]
For this series, that translates to:
\[
a_n = 3 \cdot 3^{n-1} = 3^n
\]
We can represent the sum of this series in sigma notation as follows:
\[
\sum_{n=1}^{5} 3 \cdot 3^{n-1}
\]
Therefore, the correct option for the sigma notation of the finite geometric series from the given sequence is:
\[
\sum_{n=1}^{5} 3 \cdot 3^{n-1}
\]
Among the responses you provided, the correct answer is:
\(\sum_{n=1}^{5} 3(3)^{n-1}\) (which matches the calculated expression).