Sounds like the area between every 2nd consecutive circle is shaded.
Therefore my suggestion is
$\displaystyle Area = (\pi.48^2-\pi.47^2)+(\pi.46^2-\pi.45^2)+\cdots+(\pi.2^2-\pi.1^2) $
$\displaystyle Area = \pi[(48^2-47^2)+(46^2-45^2)+\cdots+(2^2-1^2)] $
Should be fairly easy from there!
Hello, aaaaaa1234!
A circle has area: .$\displaystyle \pi r^2$48 concentric circles are drawn having radii 1, 2, 3, ... 48.
The area outside the circles of odd radii but inside the circles of even radii are shaded.
What is the area of the shaded region?
The shaded area is: .$\displaystyle A \:=\:(\pi\!\cdot\!2^2 - \pi\!\cdot\!1^2) + (\pi\!\cdot\!4^2 - \pi\!\cdot\!3^2) + (\pi\!\cdot\!6^2-\pi\!\cdot\!5^2) + \hdots + (\pi\!\cdot\!48^2 - \pi\!\cdot\!47^2)$
. . $\displaystyle = \;3\pi + 7\pi + 11\pi + \hdots + 95\pi \;=\; \underbrace{(3 + 7 + 11 + \hdots + 95)}_{\text{arithmetic series}}\pi $
The series has: first term $\displaystyle a = 3$, common difference $\displaystyle d = 4$, and $\displaystyle n = 24$ terms.
. . Its sum is: .$\displaystyle \tfrac{24}{2}[2(3) + 23(4)] \:=\:1176$
Therefore: .$\displaystyle A \;=\;1176\pi$
When visualising the problem you should see similar to an achery target.
Every second circle (or hoop section) coloured in. My equation calculates the area of each hoop (as the difference of the area of the outer part and inner part of each coloured circle) and summing them together to find total area. The area of the first hoop (the biggest hoop) is simply
$\displaystyle Area = (\pi.48^2-\pi.47^2) $
then the next biggest
$\displaystyle Area = (\pi.46^2-\pi.45^2) $
then the next biggest
$\displaystyle Area = (\pi.44^2-\pi.43^2) $
and so on until you get to the end
$\displaystyle Area = (\pi.2^2-\pi.1^2) $
Add all these guys together and you get
$\displaystyle Total Area = (\pi.48^2-\pi.47^2)+(\pi.46^2-\pi.45^2)+\cdots+(\pi.2^2-\pi.1^2) $
In the next line of my calculation I simply removed $\displaystyle \pi $ as a common factor.
$\displaystyle Total Area = \pi[(48^2-47^2)+(46^2-45^2)+\cdots+(2^2-1^2)] $
I arrived at the solution $\displaystyle Total Area = \pi.1176 \approx 3692.64 units^2 $
No that arithmetic isn't very robust.
$\displaystyle 48^2-47^2 \neq (48-47)^2 $
The arithmetic you should use is.
$\displaystyle 48^2 = 2304 $
$\displaystyle 47^2 = 2209 $
So
$\displaystyle 48^2-47^2 = 2304-2209 = 95$
You would have to repeat that calculation for every hoop before summing together.
I would let ms excel do the work for me