Answer
proof by mathematical induction provided below
Work Step by Step
proof by mathematical induction:
suppose P(n) = 1 + 3n ≤ 4^n, for every integer n ≥ 0.
Basis step: Show that P(0) is true: P(0) = 1+ 3(0) = 1 ≤ 1 (True)
Inductive Step: Show that for all integers K ≥ 0, if P(K) is true then P(K+1) is true:
suppose P(K) = 1+3K ≤ 4^K is true (Inductive hypothesis)
P(K+1) = 1 + 3(k+1) ≤ 4^(K+1) = 1 + 3k + 3 ≤ 4 (4^K)
but 1 + 3k is ≤ 4^k ( by P(K) ),
so it yields: (1 + 3K) + 3 ≤ 4^k + 3 (4^K)
but 3 ≤ 3 (4^K) for all K ≥ 0
so (1 + 3K) + 3 ≤ 4^(K+1) is true
therefore, P(n) is true for all n ≥ 0