i need to prove the following:
let:
a1 = 3
an+1 = (1/3)(an + 12)
for all n in N. show that the sequence an converges, and then find the limit
Thanks!
You need to first show that the sequence is bounded from above [i.e. prove by induction that $\displaystyle a_k<M\,(M\in\mathbb{R})$] and show that it is monotone [prove by induction that $\displaystyle s_{k}<s_{k+1}$]. Once you've done that, note that as $\displaystyle n\to\infty$, $\displaystyle a_{n+1}\to a,\,a_{n}\to a$. Thus $\displaystyle a=\tfrac{1}{3}\left(a+12\right)\implies a=\dots$
Can you take it from here?