Question

Prove by mathematical induction 2^n < 3^n

Answer 1

We know,

If a < b and c < d, then ac < bd.

Let n = 1.

2¹ < 3¹ => 2 < 3

Let n = 2.

2² < 3² => 4 < 9

Let n = 3.

2³ < 3³ => 8 < 27

Let n = 4.

2⁴ < 3⁴ => 16 < 81

Let n = k.

Assume that 2^k < 3^k.

Let n = k + 1.

Consider 2^(k + 1) < 3^(k + 1).

2^(k + 1) < 3^(k + 1)

=> 2^k • 2 < 3^k • 3

Here, we assumed earlier that 2^k < 3^k. To this, 2 and 3 are multiplied to LHS and RHS respectively. Hence, according to the above concept, we can say,

2ⁿ < 3ⁿ for any positive integer n.

Hence Proved!