Answer
proof by Mathematical induction provided below
Work Step by Step
proof by mathematical induction:
suppose P(n) = 5^n + 9 < 6^n, for every integer n ≥ 2.
Basis step: Show that P(2) is true: P(2) = 5^2+ 9 = 25 + 9 = 34 < 6^2 = 36 (True)
Inductive Step: Show that for all integers K ≥ 2, if P(K) is true then P(K+1) is true:
suppose P(K) = 5^k + 9 < 6^K is true (Inductive hypothesis)
P(K+1) = 5^(K+1) + 9 < 6^(K+1)
5 ( 5^K ) + 9 < 6 (6^K)
4 (5^K) + (5^K + 9) < 5(6^K) + (6^K)
comparing both sides: 5^K + 9 is less than 6^k by the inductive hypothesis P(K)
and its obvious that 4 (5^K) is less than 5(6^K) for all integers K >= 2 [can be proved same way by Mathmematical Induction]
so 5^(K+1) + 9 is less than 6^(K+1), and therefore P(n) os true for all integers n greater than or equals 2.
