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
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'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'
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.