Prove by Induction 1 2n n3 for all n 2SolutionPn 12n n3

Prove by Induction:

1 + 2n < n³ for all n >= 2

Solution

P(n); 1+2n < n³ , n>=2

Basic Step :

Put n =2,

L.H.S,

1+2(n) = 1+4=5

R.H.S.

n³=8

5<8

Hence Proved,

Now Let us assume that P(k) is true and we can prove that P(k+1) is also true,

P(k): 1+2(k)<k³.....................(1)

P(k+1): 1+2(k+1)<(k+1)³

1+2k+2< k³+1+3k²+3k

k³>1+3k²+3k

and We know that k<k², 1<k

So, we can write it as in in this form

3k²+3k+1< 3(k²)+3(k²)+1(k²)

7k²<k³

So, we can write it as

1+2(k+1)<(k+1)³

Hence,

By the principal of mathematical induction P(k+1) is true for all numbers greater than or equal to 2.