A square is drawn by joining mid pint of the sides of a square. Another square is drawn inside the second square in the same way and the process is continued in definitely. If the side of the first square is 16 cm, then what is the sum of the areas of all the squares ?

Options

A. 256 sq cm
B. 512 sq cm
C. 1024 sq cm
D. 512/3 sq cm

Explanation

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

### Step 1: Calculate the area of the first square.
The side of the first square is given as 16 cm. The area \( A_1 \) of a square is calculated using the formula:
\[ A_1 = \text{side}^2 = 16^2 = 256 \text{ cm}^2 \]

### Step 2: Determine 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 as:
\[ \text{side of second square} = \frac{\text{side of first square}}{\sqrt{2}} = \frac{16}{\sqrt{2}} = 8\sqrt{2} \text{ cm} \]

### Step 3: Calculate the area of the second square.
Using the side length found in Step 2, the area \( A_2 \) of the second square is:
\[ A_2 = \left(8\sqrt{2}\right)^2 = 64 \times 2 = 128 \text{ cm}^2 \]

### Step 4: Determine the side length of the third square.
Continuing the process, the side length of the third square is:
\[ \text{side of third square} = \frac{\text{side of second square}}{\sqrt{2}} = \frac{8\sqrt{2}}{\sqrt{2}} = 8 \text{ cm} \]

### Step 5: Calculate the area of the third square.
The area \( A_3 \) of the third square is:
\[ A_3 = 8^2 = 64 \text{ cm}^2 \]

### Step 6: Identify the pattern in the areas.
We can observe that the areas of the squares form a geometric series:
- \( A_1 = 256 \)
- \( A_2 = 128 \)
- \( A_3 = 64 \)
- Continuing this pattern, we see that each subsequent area is half of the previous area.

### Step 7: Write the sum of the areas as a geometric series.
The sum of the areas \( S \) can be expressed as:
\[ S = A_1 + A_2 + A_3 + \ldots = 256 + 128 + 64 + \ldots \]
This is a geometric series where:
- First term \( a = 256 \)
- Common ratio \( r = \frac{1}{2} \)

### Step 8: 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:
\[ S = \frac{256}{1 - \frac{1}{2}} = \frac{256}{\frac{1}{2}} = 256 \times 2 = 512 \text{ cm}^2 \]

### Final Answer:
The sum of the areas of all the squares is \( 512 \text{ cm}^2 \).

---