Answered

n ∈ lN

We set A= n(n²+11)

Show that A is divisible by 3

( n is natural number )

please try to help me in this i need it tomorrow plz...

Verified Answer :

konrad509

\(n(n^2+11)=\ (n^2-1+12)=\\ n(n^2-1)+12n=\\ (n-1)n(n+1)+12n\)

\((n-1)n(n+1)\) is the product of 3 consecutive natural numbers, so it has be to divisible by 3.

\(12n\) is also divisible by 3 because 12 is divisible by 3.

If both elements of the sum are divisible by 3 so the sum itself is divisible by 3 as well.

@Konrad509's answer was fantastic. 

What you first need to know is this. 3 consecutive natural numbers multiplied by each other, for instance: (123) or (234) or (345) or (45(23)) can be described using the abstract expression:

(n-1)n(n+1)

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Now:

A=n(n²+11),

and:

n(n²+11)

=n(n²-1+12)

=n³-n+12n

=n(n²-1)+12n

=n(n+1)(n-1)+12n

=(n-1)n(n+1)+(43)n

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So:

A=(n-1)n(n+1)+(4*3)n

Therefore, A is divisible by 3.