Respuesta :
Answer:
(n+5)² - (n+3)² =
= (n² + 10n + 25) - (n² + 6n + 9)
= n² + 10n + 25 - n² - 6n - 9
= 4n + 16
= 4(n + 4) ⋮ 4
It is Proved that the multiples of (n+5)² - (n+3)² are 4(n + 4) and 4 for all positive integer values of n.
What is the fundamental principle of multiplication?
If an event can occur in m different ways and if following it, a second event can occur in n different ways, then the two events in succession can occur in m × n different ways.
We have to Prove that (n+5)^2 - (n+3)^2 is a multiple of 4 for all positive integer values of n.
Given;
(n+5)² - (n+3)²
Expand each bracket
(n² + 10n + 25) - (n² + 6n + 9)
n² + 10n + 25 - n² - 6n - 9
Collect Like Terms;
= 4n + 16
Now, Factorize;
= 4(n + 4) ⋮ 4
Hence, Proved that the multiples of (n+5)² - (n+3)² are 4(n + 4) and 4.
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