Hall and Knight question

Question

If n is any positive integer show that the integral part of $$(3+\sqrt7)^n$$is a odd number

I have no idea how to begin this problem but it is given in the chapter of binomial theorem so I hope that it is found using that only

Answer 1

Hint:

Consider $(3+\sqrt7)^n+(3-\sqrt7)^n$

Answer 2

'I'to denote the integral and 'f'to denote the fractional part of $(3+√7)^n$

Now $(3-√7)^n$ is less than 1 and a proper fraction let's denote it by f'

$(3+√7)^n=3^n+ C_13^{n-1}√7......$

$3-√7)^n=3^n-C_13^{n-1}√7........$

As you can see when we add them the irrational terms cancel out.

$(3+√7)^n+(3-√7)^n$=I+f+f'= even integer

But since f and f' are proper fractions there are some must be 1

Hence we conclude that it's integral part is odd.