To Prove.

Asked by Sejal Gupta | 16th Sep, 2010, 05:11: PM

Expert Answer:

Dear Student,
 
The given question can be proved by Mathematical Induction.
 
For the base case, n = 1, we have, x2 - y2 = (x+y)(x-y) , which is divisible by (x-y).
 
Let us assume that the given hypothesis is true upto n = N.
i.e. x2N - y2N is divisible by x - y.
=> x2N - y2N =  k(x - y) for some integer k.
=> x2N = y2N + k(x-y)             ------ (1)
 
We need to prove that it is also true for n = N + 1.
for n = N + 1
we have,
 
x2N+2 - y2N+2
 
= x2N.x2 - y2N.y2
 
= (y2N + k(x-y)).x2 - y2N.y2                    ------- from (1).
 
= y2N.x2 + k.x2(x-y) - y2N.y2
 
= y2N(x2 - y2) + k.x2.(x-y)
 
= y2N(x - y)(x+y) + k.x2.(x-y)
 
= (x-y)(y2N(x+y) + k.x2)
 
Which is again divisible by (x-y).
 
Hence by mathematical induction we have proved that x2n - y2n is divisible by (x-y) for all integers n>=1.

Answered by  | 16th Sep, 2010, 10:10: PM

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