Question

the formula f(x+1)=2/3(f(x)) defines a geometric sequence where f(1)=18. Which explicit formula can be used to model the same sequence?

Answer 1

\[f(x+1)=\frac{2}{3}f(x)\implies f(x)=\frac{2}{3}f(x-1)\]which means
\[\begin{align*}
f(x+1)&=\frac{2}{3}f(x)\\[1ex]
&=\frac{2}{3}\left(\frac{2}{3}f(x-1)\right)=\left(\frac{2}{3}\right)^2f(x-1)\\[1ex]
&=\left(\frac{2}{3}\right)^2\left(\frac{2}{3}f(x-2)\right)=\left(\frac{2}{3}\right)^3f(x-2)\\[1ex]
&~~\vdots
\end{align*}\]Can you see the general pattern here?