NCERT Solutions
Class 11 Maths
Principle of Mathematical Induction
Ex.4.1 Q.4
Prove the following by using the principle of mathematical induction for all n є N:
1.2.3 + 2.3.4 + ........+ n (n + 1) (n + 2) = {n (n + 1) (n + 2) (n + 3)} ÷ 4
Let the given statement be P(n), i.e.,
P(n): 1.2.3 + 2.3.4 + ............ + n (n + 1) (n + 2) = {n (n + 1) (n + 2) (n + 3)} ÷ 4
For n = 1, we have
P (1): 1.2.3 = 6 = {1(1 + 1) (1 + 2) (1 + 3)} ÷ 4 = (1 × 2 × 3 × 4) ÷ 6 = 6, which is true.
Let P(k) be true for some positive integer k, i.e.,
1.2.3 + 2.3.4 + ... + k (k + 1) (k + 2) = {k (k + 1) (k + 2) (k + 3)} ÷ 4 ………. (1)
We shall now prove that P (k + 1) is true.
Consider
1.2.3 + 2.3.4 + ... + k (k + 1) (k + 2) + (k + 1) (k + 2) (k + 3)
= {1.2.3 + 2.3.4 + ... + k (k + 1) (k + 2)} + (k + 1) (k + 2) (k + 3)
= {k (k + 1) (k + 2) (k + 3)} ÷ {4 + (k + 1) (k + 2) (k + 3)}
= {(k + 1) (k + 2) (k + 3)} (
+ 1)
= {(k + 1) (k + 2) (k + 3) (k + 4)} ÷ 4
= {(k + 1) (k + 1 + 1) (k + 1 + 2) (k + 1 + 3)} ÷ 4
Thus, P (k + 1) is true whenever P(k) is true.
Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., N.