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# Find an explicit rule for the nth term of a geometric sequence where the second and fifth terms are -6 and 162, respectively.

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goat

The general formula for the nth term of a geometric sequence is: $n_{th}\ term=a_{1}r^{n-1}$ We can write the following for the second term: $-6=a_{1}r\ ............(1)$ and for the fifth term: $162=a_{1}r^{4}\ ............(2)$ Dividing equation (2) by equation (1) gives: $-\frac{162}{6}=\frac{a_{1}r^{4}}{a_{1}r}=r^{3}$ $r= \sqrt[3]{-27} =-3$ $a_{1}=\frac{-6}{-3}=2$ The required explicit rule is therefore: $n_{th}\ term=2(-3)^{n-1}$

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