Show convergence

**janae77** · Apr 17th 2010, 07:04 PM

if a $_{n+2}$ = (a $_{n+1}$ + a $_n$ )/2 for all n greater or equal to 1. show that
a $_n$ converges to (a $_n$ + 2a $_2$ )/3.

Please help me. I know that we want to show that the absolute value of a $_n$ minus(a $_n$ + 2a $_2$ )/3 is less than epsilon but I am not sure how to get there. Please help.

Ok so am i supposed to know what the first few terms are? can somebody please help me?

**HallsofIvy** · Apr 18th 2010, 05:18 AM

The question itself makes no sense. If $a_n$ is sequence indexed by "n" then its limit cannot involve "n". Are you sure you have copied this correctly?

**Laurent** · Apr 18th 2010, 06:06 AM

Originally Posted by janae77

if $a_{n+2}=\frac{ a_{n+1} + a_n}{2}$ for all n greater or equal to 1. show that
$a_n$ converges to $\frac{a_1+2a_2}{3}$ .

(I corrected the question)

You can notice that the sequence $u_n=a_{n+1}-a_{n}$ satisfies $u_{n+1}=-\frac{1}{2}u_n$ and $u_1=a_2-a_1$ hence by induction $u_{n+1} = \frac{(-1)^n}{2^n}(a_2-a_1)$ (this is a geometric sequence), and you can then compute $a_{n+1}-a_1=u_n+u_{n-1}+\cdots+u_1=\cdots$ (sum of terms from a geometric sequence) and conclude by taking the limit as $n\to\infty$ .

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