- #1

- 1

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**Homework Statement**

Consider the sequence {a

_{n}} where a

_{1}= sqrt(k), a

_{n+1}= sqrt(k + a

_{n}), and k > 0.

a. Show that {a

_{n}} is increasing and bounded.

b. Prove that the limit as n approaches infinity of a

_{n}exists.

c. Find the limit as n approaches infinity of a

_{n}.

**The attempt at a solution**

b is straightforward. If you show that a

_{n}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 a

_{n}<= ((1 + sqrt(1 + 4k))/2). I don't understand how I would be able to go from the givens to that point.