Problem 90E

90. (a) A sequence $\left\{a_{n}\right\}$ is defined recursively by the equation $a_{n}=\frac{1}{2}\left(a_{n-1}+a_{n-2}\right)$ for $n \geqslant 3$, where $a_{1}$ and $a_{2}$ can be any real numbers. Experiment with various values of $a_{1}$ and $a_{2}$ and use your calculator to guess the limit of the sequence.

(b) Find $\lim _{n \rightarrow \infty} a_{n}$ in terms of $a_{1}$ and $a_{2}$ by expressing $a_{n+1}-a_{n}$ in terms of $a_{2}-a_{1}$ and summing a series.

Step-by-Step Solution