# Sequences (Induction?) Problem

1. Jan 22, 2009

### bz89

The problem statement, all variables and given/known data
Consider the sequence {an} where a1 = sqrt(k), an+1 = sqrt(k + an), and k > 0.

a. Show that {an} is increasing and bounded.

b. Prove that the limit as n approaches infinity of an exists.

c. Find the limit as n approaches infinity of an.

The attempt at a solution

b is straightforward. If you show that an is monotonic and bounded then it has a limit.

I don't really understood how to approach a. The solutions guide suggests some sort of induction that starts with an <= ((1 + sqrt(1 + 4k))/2). I don't understand how I would be able to go from the givens to that point.

2. Jan 23, 2009

### dirk_mec1

a) calculate a(n+1)-a(n) and draw your conclusion.

b)-

c) In the limit for n tending to infinity you'll get: $$L = \sqrt{k+L}$$ which you can solve.

Last edited: Jan 23, 2009