Question

Given the two terms in a geo metric sequence find the 8th term and the recursive formula. A1 = -2 a5 = -512

Answer 1

Check the picture below.

`-2r^4=-512\implies r^4=\cfrac{-512}{-2}\implies r^4=256 \\\\\\ r=\sqrt[4]{256}\implies r=4 \\\\[-0.35em] ~\dotfill\\\\ n^(th)\textit{ term of a geometric sequence} \\\\ a_n=a_1\cdot r^(n-1)\qquad \begin{cases} a_n=n^(th)\ term\\ n=\textit{term position}\\ a_1=\textit{first term}\\ r=\textit{common ratio}\\[-0.5em] \hrulefill\\ a_1=-2\\ r=4 \end{cases}`

`{\Large \begin{array}{llll} a_n=-2\cdot 4^(n-1)\\\\ \stackrel{ \textit{explicit format} }{a_n=a_(n-1)\cdot 4} \end{array}} \hspace{9em}\stackrel{\textit{the 8th term}}{a_8=-2\cdot 4^(8-1)} \implies a_8=-32768`