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23 June, 12:04

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

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

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  1. 23 June, 13:21
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    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!
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