Question: What is the summation notation for the geometric series \(5 + 10 + 20 + 40 + 80\)? \[…

What is the summation notation for the geometric series \(5 + 10 + 20 + 40 + 80\)?

\[ \sum_{k=1}^{5} 2(5)^{k+1} \]

\[ \sum_{k=0}^{4} 5(2)^{k+1} \]

\[ \sum_{k=1}^{4} 5(2)^{k-1} \]

\[ \sum_{k=1}^{5} 5(2)^{k-1} \]

Solution

The problem is asking for the summation notation of the geometric series: \[ 5 + 10 + 20 + 40 + 80 \] To find the summation notation, identify the first term and the common ratio. 1. The first term \(a\) is 5. 2. The common ratio \(r\) can be found by dividing the second term by the first term: \[ r = \frac{10}{5} = 2 \] 3. The series has 5 terms. The general formula for the nth term of a geometric series is given by: \[ a \cdot r^{n-1} \] Thus, the series can be represented by the summation notation: \[ \sum_{k=1}^{5} 5 \cdot 2^{k-1} \] Therefore, the answer that matches this notation is: \[ \sum_{k=1}^{5} 5(2)^{k-1} \]