Let P(n) be the statement: 2 n ≥ 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?

Question

Philoid · Accepted Answer

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

So, it is not true for n=1

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