A square is drawn by joining the mid points of the sides of a square. A third square is drawn inside the second square in the same way and the process is continued indefinitely. If the side of the first square is 15 cm, then find the sum of the areas of all the squares so formed.

Answer

To solve the problem of finding the sum of the areas of all the squares formed by joining the midpoints of the sides of a square indefinitely, we can follow these steps:

### Step 1: Determine the area of the first square
The side length of the first square is given as 15 cm. The area \( A_1 \) of a square is calculated using the formula:
\[ A_1 = \text{side}^2 = 15^2 = 225 \text{ cm}^2 \]

### Step 2: Find the side length of the second square
The second square is formed by joining the midpoints of the first square. The side length of the second square can be calculated using the properties of right triangles. The side length of the second square \( s_2 \) is given by:
\[ s_2 = \frac{15}{\sqrt{2}} \text{ cm} \]
Thus, the area \( A_2 \) of the second square is:
\[ A_2 = s_2^2 = \left(\frac{15}{\sqrt{2}}\right)^2 = \frac{225}{2} \text{ cm}^2 \]

### Step 3: Find the side length of the third square
Continuing this process, the side length of the third square \( s_3 \) is:
\[ s_3 = \frac{s_2}{\sqrt{2}} = \frac{15}{\sqrt{2} \cdot \sqrt{2}} = \frac{15}{2} \text{ cm} \]
Thus, the area \( A_3 \) of the third square is:
\[ A_3 = s_3^2 = \left(\frac{15}{2}\right)^2 = \frac{225}{4} \text{ cm}^2 \]

### Step 4: Generalize the area of the n-th square
From the pattern, we can see that the area of the n-th square can be expressed as:
\[ A_n = \frac{225}{(2^{n-1})} \text{ cm}^2 \]

### Step 5: Sum of the areas of all squares
The sum of the areas of all squares can be represented as an infinite geometric series:
\[ S = A_1 + A_2 + A_3 + \ldots = 225 + \frac{225}{2} + \frac{225}{4} + \ldots \]
This series has a first term \( a = 225 \) and a common ratio \( r = \frac{1}{2} \).

### Step 6: Use the formula for the sum of an infinite geometric series
The sum \( S \) of an infinite geometric series can be calculated using the formula:
\[ S = \frac{a}{1 - r} \]
Substituting the values:
\[ S = \frac{225}{1 - \frac{1}{2}} = \frac{225}{\frac{1}{2}} = 225 \times 2 = 450 \text{ cm}^2 \]

### Final Answer
Thus, the sum of the areas of all the squares formed is:
\[ \boxed{450 \text{ cm}^2} \]