Question

If n ∈ N1- develop (n + 1) (n + 2)2- deduce that 16 divides 8n² + 24n + 16

Answer 1

Answer:

Explanation:

1 - Expanding out ( n + 1) ( n + 2) gives

$(n+1)(n+2)=n(n+2)+1(n+2)$$=n^2+2n+n+2$$=n^2+3n+2$

2 - Now, ( n + 1 ) ( n + 2) is a natural number. Therefore, if we multiply (n + 1 ) ( n + 2) by a multiply of 8 then the result will be divisible by 8.

The expression,

$8n^2+24n+16$

can be rewritten as

$8(n^2+3n+2)$

The left expression we recongise as what we find in part 1, and therefore, it is a natural number.

Meaning, 8 (n^2 + 3n + 2) is divisible by 8 because

$\frac{8n^2+24n+16}{8}=\frac{8(n^2+3n+2)}{8}=n^2+3n+2$

gives a natural number.