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Use sigma notation to express each series.

1. $8+11+14+17+20$
2. $\frac{2}{3}-1+\frac{3}{2}-\frac{9}{4}+\frac{27}{8}-\frac{81}{16}$
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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 $\sum_{n=1}^{6} a_{n}=\sum_{n=1}^{6} \frac{2}{3}\left(\frac{-3}{2}\right)^{n-1}$
by Diamond (39,769 points)

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