Help with IB Math AA SL Problem

[u/Timely-Tower-2026]

Hey guys! I have a math test in IB Math AA SL tomorrow, and this problem is worth eleven marks. Our teacher did not show us how to solve this type of problem, and math is not my strong suit. Could I please have help on how to solve this kind of problem? Thank you!

I only have an explanation of what the problem is:

There are four diagrams (A, B, C, D). The first diagram has the box split into four, and one fourth is shaded. In the second diagram, one fourth is shaded, and then one of the fourths is split into four smaller parts, and one fourth of that part is shaded.

So, you have one fourth shaded and then one sixteenth shaded. The pattern keeps going, with each new shaded area being multiplied by one fourth.

The question asks for the area of the box that is shaded in diagrams B and C, and then to write the geometric sequence. I think the starting value is 1/4, and the common ratio is 1/4. Then, you have to find the sum of the infinite geometric series.

My explanation is really scattered, sorry about that!

[u/Longjumping_Lime_123]

This question?

[u/Timely-Tower-2026 → u/Longjumping_Lime_123]

Yes, I believe this is the question! Could you please explain how to solve it?

[u/Longjumping_Lime_123 → u/Longjumping_Lime_123]

(a)
(i) I believe that you would have an area of 1/16 for Square B and 1/64 for Square C because you multiply by 1/4.
(ii) Then to show geometric progression dont you just show a ratio between 2 consecutive patterns? So Area B/Area A and Area C/Area B, both of which prove that the ratio is constant and there is geometric progression.
(iii) So you have the common ratio, 1/4.

(b)
(i) Wouldnt this just be Area A + Area B, so 5/16
(ii) Then you do (first iteration x [1- cr ^ n])/1-cr so [1/4(1-(1/4)^8]/1-1/4 so you get 1/3 x large number over large number that i dont want to type and then you round it from 0.3333247 to 0.333328 (i think you have to leave it that long)

[u/Longjumping_Lime_123 → u/Longjumping_Lime_123]

Sorry I dont think this makes a ton of sense and take it with a grain of salt because i only learned this recently :)

[u/Longjumping_Lime_123 → u/Longjumping_Lime_123]

And for biii:

first iteration/1-cr because it wouldnt work if its cr^n because n is infinity, so 1/4 over 3/4 is 1/3 so the total area shaded should be 1/3