## panchtatvam 3 years ago prove using mathematical induction \[x ^{n} - y^{n} = (x-y)(x^{n-1} + x^{n-2}y +.....+ xy^{n-2} + y^{n-1}\] I'm not even able to prove it true for n =1 . How could one reduce the term in the second bracket to 1 ?

1. experimentX

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2. panlac01

fundamental theorem of algebra

3. experimentX

n=1 is the base case

4. panchtatvam

@experimentX 1 is the base case but it also has to be proved first.

5. panchtatvam

I'm unable to reduce the expression in the second bracket as it involves inverse terms with dont cancel to 1

6. experimentX

in your formula n-1 > 0, but n-1 =0 for n=1

7. A.Avinash_Goutham

well u can do it this way.......... (x-y)(x^n+y^n+.....) (x^(n) - y^(n))(x-y)+ y(x-y)(xn−1+xn−2y+.....+xyn−2

8. experimentX

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9. A.Avinash_Goutham

@experimentX is that the final step?

10. experimentX

no .. not really, currently, i cannot think using induction.

11. A.Avinash_Goutham

no it's purely based on induction.....

12. experimentX

it proves directly using geometric sum

13. A.Avinash_Goutham

u prove very dumb things using induction/by contradiction...........

14. A.Avinash_Goutham

so now by induction.....we have xn−yn=(x−y)(xn−1+xn−2y+.....+xyn−2+yn−1 by induction

15. A.Avinash_Goutham

nd we need to prove xn+1−yn+1=(x−y)(xn+xny+.....+xyn-1+yn......

16. A.Avinash_Goutham

so try showing that (x−y)(xn+xny+.....+xyn-1+yn...... = xn+1−yn+1

17. panchtatvam

if we follow induction methods then as per @experimentX the formula is valid only for natural indexes . so n =1 gets proved . for n+1 could be proved by solving the RHS instead of adding any term to the value for the equation for n.

18. experimentX

|dw:1343195660262:dw| i guess ... certainly, other ways are more intuitive

19. experimentX

\[ (x-y)(x^n + x^{n-1}y + x^{n-2}y^2 + ... +y^n) \\ = x^n(x-y) + y (x-y)(x^{n-1}+x^{n-2}y + ...+y^{n-2}) \\ = x^n(x-y)+y (x^n - y^n) = x^{n+1} - y^{n+1}\]

20. panchtatvam

I wanted to have a mathematical Induction proof of the problem . But as it comes out the problem needs to have certain assumptions and can't be explained using mathematical induction in the normal way. Assumptions : 1. n > 1 so as shown by @experimentX we need to work the problem from RHS to LHS to prove the second condition of the induction thoerem.