Write the first five terms of the recursively defined sequence. $$s_{1}=1\qquad s_{n}=4s_{n-1}\qquad n\geq 2$$

Solution The first term is given as $s_{1}=1$. To get the second term, we use $n=2$ in the formula to obtain $s_{2}=4s_{1}=4\cdot 1=4$. To obtain the third term, we use $n=3$ in the formula, getting $s_{3}=4s_{2}=4\cdot 4=16.$ Each new term requires that we know the value of the preceding term. The first five terms are $$\begin{eqnarray*} s_{1}&=&1 \\ s_{2}&=&4s_{2-1}=4s_{1}=4\cdot 1=4 \\ s_{3}&=&4s_{3-1}=4s_{2}=4\cdot 4=16 \\ s_{4}&=&4s_{4-1}=4s_{3}=4\cdot 16=64 \\ s_{5}&=&4s_{5-1}=4s_{4}=4\cdot 64=256 \end{eqnarray*}$$