?(a) A sequence \(\left\{a_{n}\right\}\) is defined recursively by the equation

Chapter 9, Problem 94

(choose chapter or problem)

(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 \(n \geq 3\), where \(a_{1}\) and \(a_{2}\) can be any real numbers. Experiment with various values of \(a_{1}\) and \(a_{2}\) and use a 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.