Takshila Learning - Online Tutorials

Class 11 Maths NCERT Online Solutions - Binomial Theorem Ques.  If P (n) denotes 2 n > 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) : 2 1 > 2 – 1, i.e., 2 > 1 P (k) : 2 k > k – 1 P (k + 1) : 2 k+1 > (k + 1) – 1, i.e., 2 k+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 +…