Prove, by induction, that 12^n > 7^n + 5^n for all n>1.?

Answer 1

Let P(k) be the proposition that
12^k > 7^k + 5^k where k is a value of n

Must now prove that P(2) is true and P(k + 1) is true:-

Consider P(2)
LHS = 12² = 144
RHS = 49 + 25 = 74

144 > 74 thus P(2) is true

Consider P(k + 1)
12^(k + 1) >7^(k + 1) + 5^(k + 1)
have to show P(k + 1) to be true:-
Now 12^k > 7^k + 5^k is true
12^k x 12 > 12(7^k + 5^k)
12^(k + 1) > 12(7^k + 5^k) > 7^k + 5^k
Thus P(k + 1) is true

P(2) true
P(k + 1) true
Thus P(n) is true.

Answer 2

It is true for n = 2
Let it be true for n = k
=> 12^k > 7^k + 5^k
=> 12^(k + 1) > (7 + 5) * (7^k + 5^k)
=> 12^(k + 1) > 7^(k + 1) + 5^(k + 1) + 7*5^k + 5*7^k
=> 12^(k + 1) > 7^(k + 1) + 5^(k + 1)
Hence it is true for all n > 2.