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

Recall that the \(n^{\text{th}}\) term of a geometric sequence is \(\large ar^{n-1}\) (assuming that we're starting at n=1). In this case, we're told the second term is -12, so that would mean that \(\large ar^{2-1} = -12 \implies ar=-12\). Likewise, the fifth term is 768, so that would mean that \(\large ar^{6-1} = 768 \implies ar^5 = 768\). Thus, you'll want to solve the system of equations \[\large \left\{\begin{aligned}ar &= -12\\ ar^5 &= 768\end{aligned}\right.\]
Do you know how to proceed from here and solve for a and r? :-)

Answer 2

well would you multiply ar=-12 by -5 so you can cancel out r?