# What is the square root of 8 to the nearest integer?

##### 1 Answer

#### Explanation:

Note that:

#2^2 = 4 < 8 < 9 = 3^2#

Hence the (positive) square root of

We can use this proximity of the square root of

Consider a quadratic with zeros

#(x-3-sqrt(8))(x-3+sqrt(8)) = (x-3)^2-8 = x^2-6x+1#

From this quadratic, we can define a sequence of integers recursively as follows:

#{ (a_0 = 0), (a_1 = 1), (a_(n+2) = 6a_(n+1)-a_n) :}#

The first few terms are:

#0, 1, 6, 35, 204, 1189, 6930,...#

The ratio between successive terms will tend very quickly towards

So:

#sqrt(8) ~~ 6930/1189-3 = 3363/1189 ~~ 2.828427#