Let Pn be the following proposition 1 2n 3n for all n 1 Det

Let P(n) be the following proposition: 1 + 2^n 3^n for all n 1. Determine whether P(n) holds. True or False

Solution

Let P be a property of positive integers such that: 1. Basis Step: P(1) is true, and 2. Inductive Step: if P(n) is true, then P(n + 1) is true. Then P(n) is true for all positive integers

P(1) : 1+ 2 <= 3

Let P(k) holds : 1 + 2^k < = 3^k

2^k < = 3^k -1 -----(1)

For P(k+1) : 1 +2^(k+1) < = 3^(k+1)

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

2^k< = (3^(k+1) -1)/2

2^k < = (3/2)*3^k - 0.5

2^k < = (1.5)3^k - 0.5 ----(2)

Lets check whether inequality holds or not

Comparimg with terms on RHS in equality1 with RHS terms in inequality 2

we see 3^k ----> 1.3*3^k ( which is a bigger term

1 -----> 0.5 ( which is a smaller term

So, we cam see that RHS terms in inequality2 : (1.5)3^k - 0.5 > 3^k -1 in inequality1

So, Our inequality holds for k+1 also.

Hence , 1 +2^(k+1) < = 3^(k+1)

So, Option : True