Question

Докажите, что при любом натуральном n значение выражения (4n+5)^2 - (4n-1)^2 кратно 6.

Answer 1

(4n+5)²-(4n-1)²=(4n+5-4n+1)(4n+5+4n-1)=6*(8n-4).

Answer 2

Proof that the expression (4n + 5)^2 - (4n - 1)^2 is divisible by 6 for any natural number n.

To prove that the expression (4n + 5)^2 - (4n - 1)^2 is divisible by 6 for any natural number n, we can simplify the expression and then analyze its divisibility by 6.

Let's start by expanding the expression:

(4n + 5)^2 - (4n - 1)^2

Expanding the squares, we get:

(16n^2 + 40n + 25) - (16n^2 - 8n + 1)

Simplifying further, we have:

16n^2 + 40n + 25 - 16n^2 + 8n - 1

Combining like terms, we get:

48n + 24

Now, let's analyze the expression 48n + 24 for divisibility by 6.

We can rewrite 48n + 24 as 6(8n + 4). This shows that the expression is a multiple of 6, as it can be expressed as the product of 6 and an integer (8n + 4).

Therefore, we have proven that the expression (4n + 5)^2 - (4n - 1)^2 is divisible by 6 for any natural number n.