1a) Define $\displaystyle (x_n)$ as being the sequence:

$\displaystyle x_1 = 3$

$\displaystyle x_{n+1} = \frac{1}{2}\cdot (x_n + \frac{3}{x_n})$

Prove $\displaystyle x_n$ converges and find the lim.

b) Let $\displaystyle b > 1$ and define $\displaystyle (x_n)$ as being the sequence:

$\displaystyle x_1 = b$

$\displaystyle x_(n+1) = \frac{1}{2}\cdot (x_n + \frac{b}{x_n})$

Prove $\displaystyle x_n$ converges and find the lim.

c) Suppose you're on an island with only a solar-powered very basic calculator. Use the result from part b to approximate $\displaystyle sqrt(17)$