Find the next five terms of each of the following sequences given by :
$ a_{1}=-1, a_{n}=\frac{a_{n-1}}{n}, n \geq 2 $

Given:

\( a_{1}=-1, a_{n}=\frac{a_{n-1}}{n}, n \geq 2 \)

To do:

We have to find the next five terms of the given sequence.

Solution: 

The next five terms in the sequence are obtained by substituting $n=2, 3, 4, 5, 6$ respectively.

When $n=2$,

$a_2=\frac{a_{2-1}}{2}$

$=\frac{a_1}{2}$

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

When $n=3$,

$a_3=\frac{a_{3-1}}{3}$

$=\frac{a_2}{3}$

$=\frac{\frac{-1}{2}}{3}$

$=\frac{-1}{2\times3}$

$=\frac{-1}{6}$

When $n=4$,

$a_4=\frac{a_{4-1}}{4}$

$=\frac{a_3}{4}$

$=\frac{\frac{-1}{6}}{4}$

$=\frac{-1}{6\times4}$

$=\frac{-1}{24}$

When $n=5$,

$a_5=\frac{a_{5-1}}{5}$

$=\frac{a_4}{5}$

$=\frac{\frac{-1}{24}}{5}$

$=\frac{-1}{24\times5}$

$=\frac{-1}{120}$

When $n=6$,

$a_6=\frac{a_{6-1}}{6}$

$=\frac{a_5}{6}$

$=\frac{\frac{-1}{120}}{6}$

$=\frac{-1}{120\times6}$

$=\frac{-1}{720}$

Therefore, the next five terms of the given sequence are $\frac{-1}{2}, \frac{-1}{6}, \frac{-1}{24}, \frac{-1}{120}$ and $\frac{-1}{720}$. 

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Updated on: 10-Oct-2022

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