how ? The square root of 2 is 1.141

Question

anonymous · Answer

It's not. It's about 1.414.

anonymous · Answer

what is the appropriate method to get that answer?

anonymous · Answer

Start with a number $$a_{0} = 1$$
Use the recursive formula $$a_{n+1} = a_n / 2 + 1 / a_n$$
For each iteration, you will get a better approximation:
$$a_1 = 1 / 2 + 1 / 1 = 3 / 2$$
$$a_2 = (3 / 2) / 2 + 1 / (3 / 2) = 3 / 4 + 2 / 3 = 17 / 2$$
$$a_3 = (17 / 12) / 2 + 1 / (17 / 12) = 17 / 24 + 12 / 17 = 577 / 408$$

anonymous · Answer

$$a_3 = 577 / 408 \approx 1.414$$

anonymous · Answer

Too see that the formula is reasonable, assume that a_n converges and show that its square converges to 2. To do this, substitute a for a_n and a_(n+1) in the formula, multiply by 2a, substract a^2 and you'll arrive at a^2 = 2.

ganeshie8 · Answer

nice :)

anonymous · Answer

thanks man