Prove the following by using the principle of mathematical induction for all n ∈ N :
x²ⁿ - y²ⁿ is divisible by x + y

Solution:

We can write

P (n) : x²ⁿ - y²ⁿ is divisible by x + y

We note that

P (1) : x^2.1 - y^2.1 is divisible by x + y = x² - y² = (x + y)(x - y), which is divisible by x + y.

Thus P (n) is true for n = 1

Let P (k) be true for some natural number k.

i.e., P (k) : x^2k - y^2k is divisible by x + y

We can write

x^2k - y^2k = a (x + y) .... (1)

where a ∈ N .

Now, we will prove that P (k + 1) is true whenever P (k) is true.

Now,

x^{2(k + 1)} - y^{2(k + 1)}

x^{2k + 2} - y^{2k + 2}

= x² (x^2k) - y² (y^2k)

= x² (x^2k - y^2k + y^2k) - y² (y^2k) (added and subtracted y^2k)

= x² (x^2k - y^2k) + x² y^2k - y² (y^2k)

= x².a (x + y) + y^2k (x² - y²)

= x².a ( x + y) + y^2k (x + y)( x - y)

= ( x + y ) [ax² + ( x - y ) y^2k], which is divisible by x + y

Thus P (k + 1) is true, whenever P (k) is true.

Hence, from the principle of mathematical induction, the statement P (n) is true for all natural numbers i.e., n ∈ N .

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NCERT Solutions Class 11 Maths Chapter 4 Exercise 4.1 Question 21

Prove the following by using the principle of mathematical induction for all n ∈ N : x²ⁿ - y²ⁿ is divisible by x + y

Summary:

We have proved that x²ⁿ - y²ⁿ is divisible by x + y by using the principle of mathematical induction for all n ∈ N