Question

Question 21 Prove the following by using the principle of mathematical induction for all n∈N: x^2n – y^2n is divisible by x + y.

Class X1 - Maths -Principle of Mathematical Induction Page 95

Answer 1

x^(2n) - y^(2n) is divisible by (x+y)
Let p(n): x^(2n)-y^(2n) is divisible by (x+y)

step1:- for n =1
P(1):x² -y² = (x - y)(x + y)
it is true .

step2:- for n= k
P(k): x^(2k)-y^(2k) is divisible by (x + y).
Let x^(2k)-y^(2k) = L(x + y) ----(1)

step3:- for n=k+1
P(k+1):x^(2k+2)-y^(2k+2)
= x^2k.x² - y^2k.y²
= {y^2k + L(x+y)}x²-y^2k.y²
= y^2k.x² + L(x + y).x²-y^2k.y²
= L(x+y)x² + y^2k(x²-y²)
=L(x+y).x² + y^2k(x -y)(x+y)
=(x+y)[Lx² + y^2k(x-y)]
which is divisible by (x + y) .
P(k+1) is true when p(k) is true.
Hence, from the principle of mathematical induction, statement is true.