2010/026) Strong induction problem

Suppose x is a real number, x does NOT equal 0, and (x + 1/x) is an integer

Prove for all n ≥ 1, x^n + 1/(x^n) is an integer.
x+ 1/x is in integer = n (given)

so (x+ 1/x)^2 = n^2

x^2 + 1/x^2 = n^2-2 is an integer
so true for 1 => true for 2

let it be true for upto k
now x^k+ 1/x^k

(x^k+ 1/x^k)(x+ 1/x)

= x^(k+1) + 1/x^(k-1) + x^(k-1) + 1/x^(k+1)

so x^(k+1) + 1/x^(k+1) = (x^k+1/x^k)(x+1/x) - (x^(k-1) + 1/x^(k-1))

if it is true for all n upto k so RHS is integer then LHS is integer so for k+ 1 so the induction step is proved

hence proved