To infinity and beyond

$\text{If you observe the figure properly , you can see that the area of each }$ $\text{square is 1/2 times of the area of the previous square.}$

$\text{Let area of the larger square be A}$

$\text{Clearly the sum of the areas will be a GP like this : }$

$S+\frac{S}{2}+\frac{S}{4}+.....$

$\text{i.e,}$

$S \times (1+\frac{1}{2}+\frac{1}{4}+.....)$

$\text{Now, the formula for infinite series of GP can be used }$

$S\times\frac{1}{1-\frac{1}{2}}$

$\text{Which is nothing but ,}$

$2\times S$

$\text{Area of larger square S is,}$

$S = 10\times 10 = 100$

$\therefore \text{Sum of the area of all the squares is,}$

$2\times 100 = \boxed{200}$

Hint: Its a G.P.(Geometric Progression).

Use sum $=\frac{a}{1-r}$

Where:

$a$ = first term of your GP.

$r$ = common ratio.

For every succeeding side, you divide that side by sq. root of 2. Therefore, from 10, next side will be 10/(sq. of 2), then 5, then so on. If the ratio for each side is sq. of 2 and the formula for the equation of a square is s^2, then the ratio for its succeeding area is 1/(sq. of 2)^2, or simply 1/2. So, the first area is 100, then 50, 25, so on. Using the formula for infinite geometric series,

S = A1 / 1-r

S = 100 / 1-(1/2)

S = 200

The next square has half the area of the previous one. We get the geometric series (with limit 2) times 100. Hence, the absolute area is 200.

It is 200 square units. The first square area is 10*10 = 100 square unit. The inner square of a square is a half size of the square. If you use a second piece of paper, cut it and keep a half, then cut the other half... continue doing it until you can no longer cut it in half, you would have the whole second piece of paper. Thus, the sum of the areas of all the squares is 200 square units.