Question

prove by mathematical induction 2^n<3^n

Answer 1

Answer:

Mathematical Induction works like this: Suppose you want to prove a theorem in the form "For all integers n greater than equal to a, P(n) is true". P(n) must be an assertion that we wish to be true for all n = a, a+1, ...; like a formula. You first verify the initial step. That is, you must verify that P(a) is true.

Answer 2

Answer:

Consider the statement P(n) to be compared between the magnitudes of 2^n and 3^n

Clearly P(1) is true since 3>2

Now,

Consider for some n=k that,

3^k > 2^k.....(1)

To suffice the proof by principle of mathematical induction we need to now prove the inequality 3^k+1>2^k+1 from equation 1

So,

3^k > 2^k

Multiplying 3 on both sides

3^k+1 > 3*2^k

=> 3^k+1 > (2+1)2^k = 2^k+1 + 2^k

Hence clearly, 3^k+1 > 2^k+1

and therefore by principle of mathematical induction

P(n) is true for all n € N.

Hope this helps you !