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

(5n+1) ^2 - (2n-1) ^2 is divisible by 7

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!