Question

Find an explicit rule for the nth term of the sequence. The second and fifth terms of a geometric sequence are 9 and 243, respectively.

Answer 1

Check the picture below.

`243=9(r)(r)(r)\implies 243=9r^3\implies \cfrac{243}{9}=r^3\implies \sqrt[3]{\cfrac{243}{9}}=r\implies \boxed{3=r} \\\\[-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=\stackrel{\textit{term position}}{5}\\ a_1=\textit{first term}\\ r=\stackrel{\textit{common ratio}}{3} \end{cases}\qquad \implies a_5=a_1\cdot 3^(5-1) \\\\\\ 243=a_1\cdot 3^(5-1)\implies 243=a_1\cdot 3^4\implies 243=a_1\cdot 81 \\\\\\ \cfrac{243}{81}=a_1\implies \boxed{3=a_1}\hspace{12em} {\Large \begin{array}{llll} a_n=3(3^(n-1)) \end{array}}`