Question

Let P(n) be the statement: 2ⁿ ≥ 3n. If P(r) is true, show that P(r + 1) is true. Do you conclude that P(n) is true for all n ϵ N?

Answer 1

If P(r) is true then 2^r ≥ 3r 

For, P(r+1) 

2^r+1 = 2.2^r 

For, x>3, 2x>x+3 

So, 2.2^r > 2^r + 3 for r >1 

⇒ 2^r+1>2^r+3 for r>1 

⇒ 2^r+1 > 3r +3 for r>1 

⇒ 2^r+1 > 3(r+1) for r>1 

So, if P(r) is true, then P(r+1) is also true. 

For, n =1, P(1): 

L.H.S = 2 

R.H.S = 3 

As L.H.S < R.H.S

So, it is not true for n = 1 

Hence, P(n) is not true for all natural numbers.