To express the given series using sigma notation, we need to identify the general term of the series and then use that term within the sigma notation.
The given series is:
[tex]\[ 7 - \frac{7}{3} + \frac{7}{9} - \frac{7}{27} \cdots \][/tex]
First, we notice that each term in the series includes the number [tex]\( 7 \)[/tex] and a power of [tex]\( \left(-\frac{1}{3}\right) \)[/tex].
Series pattern:
- The first term is [tex]\( 7 \)[/tex].
- The second term is [tex]\( -\frac{7}{3} \)[/tex] which is [tex]\( 7 \left(-\frac{1}{3}\right) \)[/tex].
- The third term is [tex]\( \frac{7}{9} \)[/tex] which is [tex]\( 7 \left(-\frac{1}{3}\right)^2 \)[/tex].
- The fourth term is [tex]\( -\frac{7}{27} \)[/tex] which is [tex]\( 7 \left(-\frac{1}{3}\right)^3 \)[/tex].
Observing this pattern, the general term seems to be [tex]\( 7 \left(-\frac{1}{3}\right)^{k-1} \)[/tex].
Thus, the series can be represented in sigma notation as:
[tex]\[
\sum_{k=1}^{12} 7 \left(-\frac{1}{3}\right)^{k-1}
\][/tex]
So, the correct expression to complete the sigma notation is [tex]\( 7 \left(-\frac{1}{3}\right)^{k-1} \)[/tex].
Final answer in sigma notation:
[tex]\[
\sum_{k=1}^{12} 7 \left(-\frac{1}{3}\right)^{k-1}
\][/tex]
