Question

Find an explicit rule for the nth term of a geometric sequence where the second and fifth terms are -12 and 768, respectively.

Answer 1

The nth term is :
\[u_n=aq^n\]
And we have :
\[u_{2}=-12~~~u_5=768\]
So we get :
\[-12=aq^2~~~(1)\\768=aq^5~~~~(2)\]
By dividing (2) over (1) we get :
\[q^3=\frac{-768}{12}=-64\]
So :
\[q=\sqrt[3]{-64}=-4\]
From (1) we have :
\[12=aq^2\]
So :
\[a=\frac{q^2}{12}=\frac{(-4)^2}{12}=\frac{16}{12}=\frac43\]
So the nth term is :
\[u_n=\frac43\times(-4)^n\]

Answer 2

Thank you so much!