Succession of geometric shapes

Question

A succession of geometric shapes is obtained by dividing squares into smaller squares. The first three geometric shapes of these successions are illustrated as follows in the figure:

Considering that the largest square of each geometric shape has an area equal to one square meter, the area painted in black of the tenth geometric shape of this sequence, expressed in square meters, approximately, is equal to:

Attempt: Reason = $\frac{1}{2}$

Sides of squares = $\frac{1}{2}, \frac{1}{4}, \frac{1}{8},... \frac{1}{1024}$

tenth searched area = $(\frac{1}{1024})^2$

Answer 1

It's right!

Then the answer should be $(\frac{1}{4})^{10} = \frac{1}{1048576}$

Note: I use $\frac{1}{4}$ straight away as the area.

Answer 2

I am assuming that the "largest square" is the outer square, so that the first painted area corresponds to $0.25 \, m^2$.

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$A(i) = \frac{1}{4} \; \text{for }i = 1$

$A(i) = \frac{1}{4} \cdot A(i-1) \; \text{for } i > 1$

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$A(i) = {\left(\frac{1}{4}\right)}^i$

$A(10) = {\left(\frac{1}{4}\right)}^{10} = {\left(\frac{1}{1024}\right)}^2 = \boxed{\frac{1}{1048576} \, m^2} \approx 1 \, mm^2$