Question

The first three terms of a sequence are given. Round to the nearest thousandth

Answer 1

`2~~,~~\stackrel{2\cdot (5)/(2)}{5}~~\stackrel{5\cdot (5)/(2)}{\cfrac{25}{2}}~~,~~...\hspace{5em}\stackrel{\textit{common ratio}}{r=\cfrac{5}{2}} \\\\[-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\\ n=9\\ r=(5)/(2) \end{cases}`

`a_9=2\left( (5)/(2) \right)^(9-1)\implies a_9=2\left( (5)/(2) \right)^8\implies a_9=2\left( \cfrac{390625}{256} \right) \\\\\\ a_9=\cfrac{390625}{128}\implies {\Large \begin{array}{llll} a_9=3051(97)/(128) \end{array}}`