# Thread: Mathematical Induction Problem

1. ## Mathematical Induction Problem

Three parts to this question:
Consider the sequence {$\displaystyle {a_n}$}, where $\displaystyle a_n = \sqrt{2+\sqrt{2+...+\sqrt{2}}}$

Use Induction to prove that {$\displaystyle {a_n}$} is an increasing sequence, that is $\displaystyle a_{n+1} > a_n$ for all n = 1, 2, 3...

Use Induction to prove that {$\displaystyle a_n$} is bounded from above by 2, that is, $\displaystyle a_n < 2$ for all n = 1, 2, 3...

Is {$\displaystyle a_n$} convergent?

2. Originally Posted by xxlvh
Three parts to this question:
Consider the sequence {$\displaystyle {a_n}$}, where $\displaystyle a_n = \sqrt{2+\sqrt{2+...+\sqrt{2}}}$

Use Induction to prove that {$\displaystyle {a_n}$} is an increasing sequence, that is $\displaystyle a_{n+1} > a_n$ for all n = 1, 2, 3...

Use Induction to prove that {$\displaystyle a_n$} is bounded from above by 2, that is, $\displaystyle a_n < 2$ for all n = 1, 2, 3...

Is {$\displaystyle a_n$} convergent?
Please write your original sequence properly, in terms of $\displaystyle n$. At the moment you just have a constant.

3. ? this is exactly how it appeared on my assignment sheet
I'm assuming that the number of roots represents "n"