NCERT Solutions
Class 11 Maths
Principle of Mathematical Induction
Ex.4.1 Q.21
Prove the following by using the principle of mathematical induction for all n ∈ N:
x2n – y2n is divisible by x + y.
Let the given statement be P(n), i.e.,
P(n): x2n – y2n is divisible by x + y
It can be observed that P(n) is true for n = 1.
This is so because x2 × 1 – y2 × 1 = x2 – y2 = (x + y) (x – y) is divisible by (x + y).
Let P(k) be true for some positive integer k, i.e.,
x2k – y2k is divisible by x + y.
Let x2k – y2k = m (x + y), where m ∈ N ..........(1)
We shall now prove that P (k + 1) is true whenever P(k) is true.
Consider
x2(k + 1) – y2(k + 1)
= x2k × x2 – y2k × y2
= x2k × x2 – y2k × x2 + y2k × x2 - y2k × y2
= x2(x2k – y2k + y2k) - y2k × y2
= x2{m (x + y) + y2k} - y2k × y2
[From equation 1]
= m (x + y) x2 + y2k × x2 - y2k × y2
= m (x + y) x2 + y2k (x2 - y2)
= m (x + y) x2 + y2k (x - y) (x + y)
= (x + y) {mx2 + y2k (x - y)}, which is a factor of (x + y).
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.