What is the area of the shaded region

For this case we must find the area of the octagon and subtract the area from the square, in order to obtain the area of the shaded region.

So:

Where:

P: It is the perimeter

a: It is the apothem

We have to:

Substituting:

On the other hand:

Finally:

Option A

, and

for

step-by-step explanation:

we can analyze the pattern at which the terms increase. let us denote , , , so that we can call

note the following:

the pattern: to get the next term value, we just add the next term number (e.g. 1st, 2nd, to the previous term value (term values are the given numbers in the question, 1, 3, 6, 10, 15,

from , to get to , we add

from , to get to , we add

etc.

we need to form a recursive formula, a formula that has the next term as a function of the previous term. so a possible recursive formula can be

, and for