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Ierofanga [76]
3 years ago
14

A) show that $n(2n + 1)(7n + 1)$ is divisible by 6 for all integers $n$.

Mathematics
1 answer:
goldenfox [79]3 years ago
8 0
Hi,

A)
\dfrac{n(2n+1)(7n+1)}{6} \\

= \dfrac{14n^3+9n^2+n}{6} \\

= \dfrac{12n^3+6n^2}{6} + \dfrac{2n^3+3n^2+n}{6} \\

=2n^3+n^2+ \dfrac{n(2n^2+3n+1)}{6} \\

= 2n^3+n^2+ \dfrac{n(n+1)(2n+1)}{6} \\

= 2n^3+n^2+ 1^2+2^2+3^2+...+n^2\ is\ an\ integer.


B)
Only if n=4*k or n=4*k+1 , k beeing an integer.

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