Writing the Terms of a Recursively Defined Sequence

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*}