x 3 n + y 3 n is divisible by x + y , if

Step-by-step solution

Let A = x 3 n + y 3 n Put n = 0 , then A = x 3 0 + y 3 0 = x + y Hence, it is divisible by x + y . Let this be true for some natural number n = k , then A = x 3 k + y 3 k is divisible by x + y ⇒ x 3 k + y 3 k = m ( x + y ) , m ∈ Z   . . . ( 1 ) . Now, for n = k + 1 , A = x 3 k + 1 + y 3 k + 1 = x 3 k · 3 + y 3 k · 3 = x 3 k 3 + y 3 k 3 = x 3 k + y 3 k x 3 k 2 + y 3 k 2 - x 3 k · y 3 k (using a 3 + b 3 = a + b a 2 - a b + b 2 = m ( x + y ) x 3 k 2 + y 3 k 2 - x 3 k · y 3 k (using