Chapter 5.2, Problem 131E

[T] Suppose that $a_1=s_1=1$ and that, for given numbers S>1 and 0 < k < I. one defines $a_{n+1}=k\left(S-S_n\right)$and $S_{n+1}=a_{n+1}+S_n$. Does $S_n$converge? If so, to what? (Hint: First argue that $S_n<S$for all n and $S_n$is increasing.)