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“At each stage of the process, what proportion of the image is coloured blue?”

I will refer to my solution to “Diminishing Returns” to answer this question (the side lengths of the inner squares):
Side length of outer square = 1
Stage 1: 0, no part of the square is blue
Stage 2: Side length of innermost blue square = √2/2, blue area = (√2/2)^2 = 2/4 of square
Stage 3: Side length of innermost purple square = ½, purple area = (½)^2 = ¼, total blue area = 2/4 -¼ = ¼ of square
Stage 4: Side length of innermost blue square = √2/4, blue area = (√2/4)^2 = 2/16 of square.
Stage 5: Side length of innermost purple square = 1/4, purple area = (1/4)^2 = 1/16, total blue area = ¼ + (2/16 - 1/16) = ¼ + 1/16

If we keep going on we will see that the area of the blue squares keep adding up in the form ∑(¼)^n.

“What would happen if we continued the process forever?”

As we have noticed, with every second stage the proportion of the square that is blue increases by (¼)^n where n is the half of the number of stages. Being a geometric series with a common ratio <1, the summation will tend to a certain value. I calculated this summation in the answer to “Diminishing Returns”,
“Now, we can finally add all these areas up to infinity and that will give us the area of the blue squares, and the fraction of the large square that is blue.

∑(¼)^n , Starting from n=1 to infinity.

Formula of a geometric sequence going up to infinity is a(1/1-r) where a is the first term of the series and r is the common ratio.

Therefore, the series sums up to (¼)*(1/(1-(¼))) = (¼)*(1/(¾)) = (¼)*(4/3) = ⅓ .

So, a third of the large square is blue.”

Now, for the last 4 patterns. Before I start working on these, I will point out that it is not stated in the question what size are the squares are divided into (eg, in the third one it does not state that the square is divided into thirds, and the next one is divided into 3 parts again), but I assume so as the answers turn out to be neater and easier to find.

In the first image, the square is divided up into 2 halves, one is blue and the other is divided up into 2 halves again, one of the quarters is pink and the other contains a repetition of the pattern. This pattern repeats infinitely. So we can think of the proportion of the square that is blue as: (½)+(½)(¼) + (½)(¼)^2 + …. , or in words, a half, plus a half of a quarter, plus a half of a quarter of a quarter, and so on. This is a geometric series, ∑(½)(¼)^n from n=0 to infinity:
= (½)∑(¼)^n
= (½)*(1/(1-(¼)))
= (½)(4/3)
= â…”
Therefore, â…” of the square is blue.

In the second image, the bottom left quarter if the square is blue, the top left and the bottom right quarters are pink, and the top right is a repetition of the pattern (if we zoom into that quarter we have the exact same pattern but with shorter side lengths)
So the proportion of the square that is blue is (¼) + (¼)(¼) + (¼)(¼)^2 + ..., or in words, a quarter, plus a quarter of a quarter, plus a quarter of a quarter of a quarter, and so on.
This geometric series is ∑(¼)(¼)^n from n=0 to infinity (I used (¼)(¼)^n rather than (¼)^(n+1) just so that the calculations are easier and I can use the last answer’s calculation).
= (¼)∑(¼)^n
= (¼)(4/3)
=â…“
Therefore, â…“ of the square is blue.

The third image shows the square divided into thirds, the middle of which is divided into 3 further parts, the left third and the upper middle ninth are blue while the left third and the lower middle ninth are pink. The central ninth is a repetition of the pattern, it’s just rotated by 180 degrees.
So the proportion of the square that is blue is (⅓ + 1/9) + (⅓ + 1/9)*(1/9) + (⅓ + 1/9)*(1/9)^2 + …, or in words, a third and a ninth, plus a third and a ninth of a ninth, plus a third and a ninth of a ninth of a ninth, and so on.
This gives us the geometric series ∑(⅓ +1/9)(1/9)^n for n=0 to infinity.
= (⅓ + 1/9)∑(1/9)^n
= (4/9)(1/1-(1/9))
= (4/9)(9/8)
= 4/8 = ½
So, ½ of the square is blue

Finally, in the fourth image, the square is divided into fifths, the 4th one from the left is blue, and the 5th one is divided up in 5 more parts, the second of the 1/25 of the square is also blue, and the top 1/25 of the square repeats the pattern.
So the proportion of the square that is blue is (⅕ + 1/25) + (⅕+1/25)(1/25) + (⅕+1/25)(1/25)^2 + …, or another way of seeing it is a third and 1/25, plus a third and 1/25 of 1/25, plus a third and 1/25 of 1/25 of 1/25, and so on.
Thus we have the geometric sequence ∑(⅓ + 1/25)(1/25)^n for n=0 to infinity.
= (⅓ + 1/25)∑(1/25)^n
= (28/75)(1/1-(1/25))
= (28/75)(25/24)
= (7/3)(â…™)
= 7/18
Therefore, 7/18 of the square is blue.

In the image that I have included, the square is divided into 2 halves with the upper one being blue, and the lower half is divided into 5 parts, the left part of which is blue. The part on the right repeats the pattern again.
So the proportion of the square that is blue is
(½ + (⅕ * ½)) + (½ + (⅕ * ½))(1/10) + (½ + (⅕ * ½))(1/10)^2 + …
On, in other words a half and a tenth, plus a half and a tenth of a tenth, plus a half and a tenth of a tenth of a tenth, and so on.
The geometric series is ∑(½ + (⅕ * ½))(1/10)^n for n=0 to infinity.
= (½ + (⅕ * ½))∑(1/10)^n
= (6/10)(1/1-(1/10))
= (6/10)(10/9)
= 6/9 = 2/3
So 2/3 of the square is blue.