Question

Use sigma notation to express each series.

1. \(8+11+14+17+20\)
2. \(\frac{2}{3}-1+\frac{3}{2}-\frac{9}{4}+\frac{27}{8}-\frac{81}{16}\)

Answer 1

1. \(8+11+14+17+20\)
This is an arithmetic series with five terms whose first term is 8 and whose common difference is 3 . Therefore, \(a_{1}=8\) and \(d=3\). The \(n\)th term of the corresponding sequence is
\[
\begin{aligned}
a_{n} &=1_{1}+(n-1) d \\
&=8+(n-1) 3 \\
&=3 n+5
\end{aligned}
\]
Since there are five terms, the given series can be written as
\[
\sum_{n=1}^{5} a_{n}=\sum_{n=1}^{5}(3 n+5)
\]

 

2. \(\frac{2}{3}-1+\frac{3}{2}-\frac{9}{4}+\frac{27}{8}-\frac{81}{16}\)
This is a geometric series with six terms whose first term is \(\frac{2}{3}\) and whose common ratio is \(-\frac{3}{2}\). Therefore, \(a_{1}=\frac{2}{3}\) and \(r=-\frac{3}{2}\). The \(n\)th term of the corresponding sequence is
\[
\begin{aligned}
&a_{n}=a_{1} r^{n-1} \\
&=\frac{2}{3}\left(\frac{-3}{2}\right)^{n-1}
\end{aligned}
\]
Since there are six terms in the given series, the sum can be written as \(\sum_{n=1}^{6} a_{n}=\sum_{n=1}^{6} \frac{2}{3}\left(\frac{-3}{2}\right)^{n-1}\)