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Home / Elementary Number Theory / For n ≥ 1,prove that n(n+1)(2n+1) / 6 is an integer

For n ≥ 1,prove that n(n+1)(2n+1) / 6 is an integer

Proof: According to the division algorithm, every n is of the form 6q, 6q + 1, 6q + 2, 6q + 3, 6q + 4, 6q + 5.Assume the first of these cases.

When n = 6q, then

n(n + 1) (2n + 1) / 6

= 6q (6q +1) (12q + 1) / 6

= q(6q + 1) (12q + 1)

which is clearly is an integer.

When n = 6q + 1, then

n(n + 1) (2n + 1) / 6

= (6q + 1)(6q + 2)(12q + 3)/6

= (6q + 1) (72q2 + 42q + 6)/6

= (6q + 1) (12q2 + 7q + 1)

is an integer in this instant also.

When n = 6q + 2, then

n(n + 1) (2n + 1) / 6

= (6q + 2)(6q + 3)(12q + 5)/6

= (36q2 + 30q + 6)(12q + 5)/6

= (6q2 + 5q + 1)(12q + 5)

is an integer.

When n = 6q + 3, then

n(n + 1) (2n + 1) / 6

= (6q + 3)( 6q + 4)(12q + 7) / 6

= (36q2 + 42q + 12)(12q + 7) /6

= (6q2 + 7q + 2)(12q + 7)

is an integer.

When n = 6q + 4, then

n(n + 1) (2n + 1) / 6

= (6q + 4)(6q + 5)(12q + 9) / 6

= (72q2 + 102q + 36)(6q + 5)/6

= (12q2 + 17q + 6)(6q + 5)

is an integer.

When n = 6q + 5, then

n(n + 1) (2n + 1) / 6

= (6q + 5)(6q + 6)(12q + 11) / 6

= (36q2 + 66q + 30)(12q + 11) / 6

= (6q2 + 11q + 5) (12q + 11)

An integer once more. Consequently, our result is established in all cases. (Proved)

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