Write the first five terms and obtain the corresponding series, whose sequence is $a_{1} = - 1, a_{n} = \frac{a_{n - 1}}{n}, n \geq 2$

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Solution

Finding sequence and corresponding series
Given, $a_{1} = - 1, a_{n} = \frac{a_{n - 1}}{n}, n \geq 2, n \in N$
Substituting $n = 2, 3, 4$ and $5$ in $a_{n}$ , we get

$a_{2} = \frac{a_{2 - 1}}{2}$
$\Rightarrow a_{2} = \frac{a_{1}}{2}$
$\Rightarrow a_{2} = \frac{- 1}{2}$

$a_{3} = \frac{a_{3 - 1}}{3}$
$\Rightarrow a_{3} = \frac{a_{2}}{3}$
$\Rightarrow a_{3} = \frac{- 1}{6}$

$a_{4} = \frac{a_{4 - 1}}{4}$
$\Rightarrow a_{4} = \frac{a_{3}}{4}$
$\Rightarrow a_{4} = \frac{- 1}{24}$

$a_{5} = \frac{a_{5 - 1}}{5}$
$\Rightarrow a_{5} = \frac{a_{4}}{5}$
$\Rightarrow a_{5} = \frac{- 1}{120}$

Hence, the first five terms of sequence are $- 1, - \frac{1}{2}, - \frac{1}{6}, - \frac{1}{24}, - \frac{1}{120}$ and the corresponding series is $(- 1) + (\frac{- 1}{2}) + (\frac{- 1}{6}) + (\frac{- 1}{24}) + (\frac{- 1}{120}) + \dots$

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