Let {an} be the sequence defined recursively by a1 = 2and an+1 = 2 + an for n 1.(a) List

Chapter 9, Problem 27

(choose chapter or problem)

I.et \(\left\{a_{n}\right\}\) he the sequence defined recursively by \(a_{1}=\sqrt{2}\) and \(a_{n+1}=\sqrt{2+a_{n}}\) for \(n \geq 1\)

(a) I.ist the first three terms of the sequence.

(b) Show that \(a_{n}<2 \text { for } n \geq 1\) .

(c) Show that \(a_{n+1}^{2}-a_{n}^{2}=\left(2-a_{n}\right)\left(1+a_{n}\right) \text { for } n \geq 1\)

(d) Use the results in parts (b) and (c) to show that \(\left\{a_{n}\right\}\) is a strictly increasing sequence. [Hint: If \(x\) and \(y\) are positive real numbers such that \(x^{2}-y^{2}>0\) , then it follows by factoring that \(x-y>0\)
(c) Show that \(\left\{a_{n}\right\}\) converges and find its limit \(L\).

Equation Transcription:

Text Transcription:

{a_n}

a_1 = sqrt 2

a_n+1 = sqrt 2 + a_n

n geq 1

a_n < 2

a_n+1 ^2 - a_n ^2 = (2-a_n) (1+a_n)

x^2 - y^2 > 0

x - y > 0