What is the area of ​​the square formed by repeatedly joining the midpoints?

The area of ​​a square is equal to the product of the side lengths of the square.

We consider a figure in which the midpoint of the sides of each square forms another square. And so on until a specific number of squares.

This graphic shows a square formed by connecting the midpoints of the squares.

For this figure, let the side length be a,

The side length of the internal square will be

L2 = (a/2)<sup>2</sup> + (a/2)<sup>2</sup>
L2 = a<sup>2</sup>(1/4 + 1/4) = a<sup>2</sup>(1/2) = a<sup>2</sup>/2
L = a<sup>2</sup>/ (\sqrt{2}).

The area of ​​square 2 = L2 = a²/2.

For the next square, area of ​​square 3 = a²/4

Let’s give an example, tge

Now we can deduce the area of ​​the continuous squares from here,

a², a²/2, a²/ 4, a²/8, …..

This is a geometric sequence with a common ratio of ½, where a² is the first term.

Example

#include <stdio.h>
#include <math.h>
int main() {
   double L = 2, n = 10;
   double firstTerm = L * L;
   double ratio = 1 / 2.0;
   double are = firstTerm * (pow(ratio, 10)) ;
   printf("The area of %lfth square is %lf", n , sum);
   return 0;
}

Output

The area of 10th square is 0.003906

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