Question

Prove that for any natural value of n the value of the expression:

(5n+1)^2 – (2n–1)^2 is divisible by 7

Answer 1

p(n)=(5n+1)^2 – (2n–1)^2=

(5n)^2 +2*(5n)*1+1^2-((2n)^2 - 2*(2n)*1+1^2)=

25n^2+10n+1-(4n^2-4n+1)=

25n^2+10n+1-4n^2+4n-1=

21n^2+14n=

7*(3n^2+2n)

7 divides p(n)=7*(3n^2+2n) for each n!