SOLUTION: prove by mathematical induction that x-y is a factor of x^n- y^n

Question 40718: prove by mathematical induction that x-y is a factor of x^n- y^n
Answer by venugopalramana(3286) (Show Source): You can put this solution on YOUR website!
prove by mathematical induction that x-y is a factor of x^n- y^n
1.test for n=1....(X^1-Y^1)=X-Y.IS DIVISIBLE BY (X-Y)
2.ASSUME IT IS TRUE FOR N=K
SO (X^K-Y^K) IS DIVISIBLE BY (X-Y)...LET X^K-Y^K=A(X-Y)...WHERE A IS AN INTEGER.
3.T.S.T FOR X=K+1....THE STATEMENT IS TRUE
X^(K+1)-Y^(K+1)=X^(K+1)-Y^(K+1)+X^K-Y^K-(X^K-Y^K)=X^K(X+1)-Y^K(Y+1)-(X^K-Y^K)
=(X^K-Y^K)(X+1-Y-1)-(X^K-Y^K)=(X^K-Y^K)(X-Y)-(X^K-Y^K)=AN INTEGER SINCE EVERY TERM IS DIVISIBLE BY X-Y...
SO IT S TRUE FOR X=K+1
4.BUT THIS IS TRUE FOR X=1..SO IT IS TRUE FOR 1+1=2...2+1=3....ETC..FOR ALL INTEGRAL VALUES OF N
RELATED QUESTIONS
Show that x - y is a factor of x^n - y^n for all positive integers n, using mathematical... (answered by math_helper,robertb)

Prove by mathematical induction that 3^(2n)-8n-1, n is a positive integer, is a multiple... (answered by Edwin McCravy)

Prove by induction that for all n (n being positive natural numbers), a) (x^n) -... (answered by stanbon,richard1234)

prove that 1/n > 1/2power of n by using mathematical... (answered by richard1234)

Use mathematical induction to prove that the statement is true for every positive integer (answered by stanbon,Edwin McCravy)

Prove a^n -1=(a-1)(a^(n-1) + a^(n-2) +.......+ a+1) by using Principle of Mathematical... (answered by KMST)

Prove n!>n^2 for n>=4 and n!>n^3 for n>=6 by using Principle of Mathematical... (answered by Edwin McCravy)

prove that (n 0) + (n 1) + (n 2) + ... + (n k) = 2^n is true using mathematical induction (answered by Shin123,ikleyn)

Prove by mathematical induction that the sum of the interior angles of a regular polygon... (answered by FrankM,ikleyn)