If , where is a natural number, prove that the integral part of is an odd integer and…

MATHEMATICS · Subjective · AoPS · Mathematics · Special Functions · Special Functions

Problem

x = (8 + 3\sqrt{7})^n

, where

n

is a natural number, prove that the integral part of

x

is an odd integer and also show that

x - x^2 + x[x] = 1

, where

[. ]

denotes the greatest integer function.