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

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