Question

PLEASE SOLVE FAST

Prove that for any natural n the value of the expression (4n+5)^2–9 is divisible by 4.

Answer 1

The expression (4n+5)²−9 is divisible by 4 for any natural number n because upon expanding, it can be expressed as a multiple of 4: 16n² + 40n + 16, which shows clear divisibility by 4.

Step-by-step explanation:

Let's prove that the expression (4n+5)²−9 is divisible by 4 for any natural number n. First, expand the expression:

(4n+5)² = (4n+5)(4n+5) = 16n² + 40n + 25

Now, subtract 9 from this result:

(4n+5)² - 9 = 16n² + 40n + 25 - 9 = 16n² + 40n + 16

Notice that both 16n² and 40n are clearly divisible by 4, since they have factors of 16 (which is 4²) and 4, respectively. Moreover, 16 is also divisible by 4. Hence:

16n² + 40n + 16 = 4(4n² + 10n + 4)

The expression is now clearly in a form that shows it is divisible by 4. Since the original expression can be written as a multiple of 4, this concludes our proof that (4n+5)²−9 is divisible by 4 for any natural number n.

Answer 2

(4n^2+10n+4)

Just Simplify (4n+5)^2-9, then divide it by 4.