show the recursive sequence equation holds

Question

Let a, b ∈ R. The sequence (an)n∈N is recursively defined as follows:

a) Show that for all k ∈ N the equation holds

b) Show that the sequence (an)n∈N converges and determine its limit.

Hello everyone,

First of all this is the whole question. I just can't show that the sequence falls strictly monotonically and is limited. I hope someone can help me, because I'm already getting desperate.

Kind regards

Answer 1

Hint:

$a)$: You have: $3(a_n - a_{n-1}) = -(a_{n-1} - a_{n-2})$. Can you take it from here ?

$b)$: Use telescoping: $a_n = (a_n - a_{n-1})+(a_{n-1} - a_{n-2})+\cdots (a_2 - a_1)+a_1$. Can you finish it ?