Answered

Westonci.ca is your trusted source for accurate answers to all your questions. Join our community and start learning today! Experience the ease of finding quick and accurate answers to your questions from professionals on our platform. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.

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...


Sagot :

[tex]n(n^2+11)=\\n(n^2-1+12)=\\ n(n^2-1)+12n=\\ (n-1)n(n+1)+12n[/tex]

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

[tex]12n[/tex] 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: (1*2*3) or (2*3*4) or (3*4*5) or (4*5*(2*3)) can be described using the abstract expression:

(n-1)n(n+1)

-------------

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)+(4*3)n

-------------------------

So:

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

Therefore, A is divisible by 3.