Question

Prove the following by using the principle of mathematical induction:

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

I tried to do, but i'm not getting. So need the steps.

Answer 1

when n=1, the given statement becomes
\[x ^{2}-y ^{2} \] is divisible by (x+y)
therefore the statement is true

Answer 2

I can give you an article on the same if you want

Answer 3

it should be true for all values of n. Not 1 alone...

Answer 4

choose a value and let me prove it

Answer 5

I need to prove it using the Principle of Mathematical Induction.

First prove it for n=1.

Then assume it for n=k.

then prove n=k+1 is true for every n=k.

Answer 6

ok

Answer 7

I see what you mean

Answer 8

Any help would be much appreciated.. as it took my most of the time.. still didn't find a comfortable way to prove it.. :(

Answer 9

is it x+y or x-y?

Answer 10

Assume it is true for k.
That is, \(x^{2k} - y^{2k}\) is divisible by (x+y) ----- (1)

Then, for k + 1, the numerator is:
\(x^{2(k+1)} - y^{2(k+1)} = x^2x^{2k} - y^2y^{2k} = x^2x^{2k} - (y^2+x^2-x^2)y^{2k} = \\
x^2x^{2k} - x^2y^{2k} - (y^2-x^2)y^{2k} = x^2(x^{2k}-y^{2k}) - (y+x)(y-x)y^{2k} \) .

The first term is divisible by (x+y) due to (1) and the second term is divisible by (x+y) due to the factor (x+y).

Therefore, if it is true for k, it is true for (k+1).

Answer 11

x+y

Answer 12

@aum Great Job.. Thanks... is this the only method??

Answer 13

For proof by induction this is what occurred to me. Others may have a different proof but I can't imagine t would be any simpler than this one as far as induction proof is concerned.

Answer 14

cleverly done xD

Answer 15

@ganeshie8 plz zhelp me sir..i have messged u