Class 11 Maths NCERT Online Solutions - Binomial Theorem

Class 11 Maths NCERT Online Solutions - Binomial Theorem

Ques. If P (n) denotes 2n> n–1, write P (1), P (k) and P (k+1), where k ∈ N.

Solution: Replacing n by 1, k and k + 1, respectively in P (n),

We get P (1) : 21> 2 – 1, i.e., 2 > 1

P (k) : 2k> k – 1

P (k + 1) : 2k+1> (k + 1) – 1, i.e., 2k+1> k

Ques. If P (n) is the statement

‘1 + 4 + 7 + (3n – 2) = n(3n − 1)/2

Write P (1), P(k) and P(k + 1).

Solution: To write P(1), the terms on the left-hand side (LHS) of P(n) continue till

3× 1 – 2, i.e., 1. So, P (1) will have only one term in its LHS, i.e., the first term.

Also, the right hand side (RHS) of P(1) = 1*(3*1-1) / 2

= 1

Therefore, P (1) is 1 = 1.

Replacing n by 2, we get

P(2) : 1 + 4 = 2*(3*2-1) / 2

i.e., 5 = 5.

Replacing n by k and k + 1, respectively, we get

P(k) : 1 + 4 + 7 + …. + (3k – 2)

=  k(3k – 1) / 2

P(k + 1) : 1 + 4 + 7 + …. + (3k – 2) + [3 (k + 1) – 2]

=(k+1)[3(k+1)-1] / 2

i.e. , 1 + 4 + 7 +…. + (3k + 1) =(k + 1)[(3k + 2) / 2.

Continue complete chapter click Binomial Theorem

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