Question

40 POINTS FOR ANYONE THAT CAN Answer

n is the middle integer of three consecutive positive integers.
The three integers are multiplied to give a product.
n is then added to the product.
Prove that the result is a cube number.
n must be used in your proof.
Your final line must consist of only,
= n^3

Answer 1

`\boxed{\textsf{ The final answer is \textbf{n}$^{\textbf{3}}$ .}}`

Explanation:

Its given that n is the middle out of the three consecutive integers . So ,

The last integer will be :-

`\sf\implies Last \ Integer \ = \ n - 1`

The next Integer will be :-

`\sf\implies Next \ Integer \ = \ n + 1`

Now the Question says that the three integers are multipled to give a product . So that would be.

`\sf\implies Product_((three\ consecutive\ integers))= (n-1)n(n+1) = (n^2-1)(n) = \pink{n^3-n}`

Now thirdly it's given that n is added to the given integer . That would be ,

`\sf\implies Adding\ n = \ n^3 - n + n = \pink{n^3}`

Here - n and +n gets cancelled. So we are ultimately left out with n³.

Hence the final number is a cube of some number.