Write the first five terms of the recursively defined sequence. $$u_{1}=1 \qquad u_{2}=1 \qquad u_{n}=u_{n-2}+u_{n-1}\quad n\geq 3$$

Solution We are given the first two terms. To obtain the third term requires that we know each of the previous two terms. $$\begin{eqnarray*} u_{1}&=&1 \\[3pt] u_{2}&=&1 \\[3pt] u_{3}&=&u_{1}+u_{2}=1+1=2 \\[3pt] u_{4}&=&u_{2}+u_{3}=1+2=3 \\[3pt] u_{5}&=&u_{3}+u_{4}=2+3=5 \end{eqnarray*}$$