Prove by induction the inequality (1 +x)ⁿ ≥ 1 + nx whenever x is positive and n is a positive integer.

(1 + x)ⁿ > = 1 + nx

P(1): 1 + x = 1 + x

P(2): (1 + x)² > 1 + 2x

P(n):

(1 + x)^{n + 1} – 1 – nx – x  = (1 + x)((1 + x)ⁿ – 1 – nx) + (1 + nx)(1 + x) – 1 – nx – x

P(n + 1) = (1 + x) P (n) + nx²

As x² is positive, therefore P(n + 1) is true