Prove by Mathematical Induction

Question

Prove by Mathematical Induction:

If x ≥ 0, then ∀n ∈N, (1+x)ⁿ ≥1 + xⁿ

Answer 1

n=1 : (1+x)^n = (1+x)^1 = 1+x = 1+x^1 = 1+x^n

n=2: (1+x)^n = (1+x)^2 = x^2 + 2x + 1 = (1 + x^2)+ 2x >= 1 + x^2 = 1+x^n, as x>=0

n=3: (1+x)^n = (1+x)^3 = x^3 + 3x^2 + 3x + 1 = (1+x^3) + (3x^2+3x) >= 1+x^3 = 1+x^n, as x>0

note that since x>=0, 2x, 3x, and 3x^2 are all non-negative

THe Induction Hypothesis:

Suppose (1+x)^n >= 1+ x^n for positive integer n

Proof:

(1+x)^(n+1) = (1+x)^n * (1+x)

>= (1+x^n)( 1+x) <--- by induction hypothesis

= 1 + x^n + x + x^n(+1)

= [1 + x^(n+1)] + x^n+x

>= 1 + x^(n+1) since x^n+x >=0 because x>=0

[end of proof]