NCERT Solutions
Class 11 Maths
Principle of Mathematical Induction
Ex.4.1 Q.19
Prove the following by using the principle of mathematical induction for all n ∈ N:
n (n + 1) (n + 5) is a multiple of 3.
Let the given statement be P(n), i.e.,
P(n): n (n + 1) (n + 5), which is a multiple of 3.
It can be noted that P(n) is true for n = 1
since 1(1 + 1) (1 + 5) = 1 × 2 × 6 = 12, which is a multiple of 3.
Let P(k) be true for some positive integer k, i.e.,
k (k + 1) (k + 5) is a multiple of 3.
So, k (k + 1) (k + 5) = 3m, where m ∈ N ........... (1)
We shall now prove that P (k + 1) is true whenever P(k) is true.
Consider
(k + 1) {(k + 1) + 1} {(k + 1) + 5}
= (k + 1) (k + 2) {(k + 5) + 1}
= (k + 1) (k + 2) (k + 5) + (k + 1) (k + 2)
= k (k + 1) (k + 5) + 2(k + 1) (k + 5) + (k + 1) (k + 2)
= 3m + (k + 1) {2(k + 5) + (k + 2)}
= 3m + (k + 1) {2k + 10 + k + 2}
= 3m + (k + 1) (3k + 12)
= 3m + 3(k + 1) (k + 4)
= 3{m + (k + 1) (k + 4)}
= 3 × q, where q = m + (k + 1) (k + 4) is some natural number.
Therefore, (k + 1) {(k + 1) + 1} {(k + 1) + 5} is a multiple of 3
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.