A square is drawn by joining the mid- points of the sides of a given square. In the same way, this process continues indefinitely. If side of the first square is 4 cm, than the sum of the area of all the squares are equal to :

Question

Accepted Answer

To determine the sum of the areas of all the squares formed by joining the midpoints of the sides of a given square, we start with the first square having a side length of 4 cm.

1. **Area of the First Square:**
   - The area of the first square is calculated as side × side = 4 cm × 4 cm = 16 cm².

2. **Area of the Second Square:**
   - The second square is formed by joining the midpoints of the sides of the first square. The side length of the second square is the diagonal of the smaller square formed by the midpoints, which is $ \frac{4}{\sqrt{2}} = 2\sqrt{2} $ cm.
   - The area of the second square is $ (2\sqrt{2})^2 = 8 $ cm².

3. **Area of the Third Square:**
   - Similarly, the third square is formed by joining the midpoints of the second square. Its side length is $ \frac{2\sqrt{2}}{\sqrt{2}} = 2 $ cm.
   - The area of the third square is $ 2^2 = 4 $ cm².

4. **Area of the Fourth Square:**
   - The fourth square is formed by joining the midpoints of the third square. Its side length is $ \frac{2}{\sqrt{2}} = \sqrt{2} $ cm.
   - The area of the fourth square is $ (\sqrt{2})^2 = 2 $ cm².

5. **Continuing the Process:**
   - This process continues indefinitely, forming a geometric series with the areas of the squares: 16, 8, 4, 2, ...

6. **Sum of the Infinite Series:**
   - The sum of the areas of all the squares forms an infinite geometric series with the first term $ a = 16 $ cm² and a common ratio $ r = \frac{1}{2} $.
   - The sum $ S $ of an infinite geometric series is given by the formula $ S = \frac{a}{1 - r} $.
   - Substituting the values, we get $ S = \frac{16}{1 - \frac{1}{2}} = \frac{16}{\frac{1}{2}} = 32 $ cm².

Therefore, the sum of the areas of all the squares is 32 cm², which corresponds to Option 1.

**Explanation for Incorrect Options:**

- **Option 2 ($16 cm^2$):** This option is incorrect because it only accounts for the area of the first square and does not include the areas of the subsequent squares formed by the process.

- **Option 3 ($64 cm^2$):** This option is incorrect because it overestimates the sum of the areas. The sum of the infinite series converges to 32 cm², not 64 cm².

- **Option 4 ($48 cm^2$):** This option is incorrect because it also overestimates the sum of the areas. The correct sum, as calculated, is 32 cm².

In conclusion, Option 1 is the correct answer as it accurately represents the sum of the areas of all the squares formed by the described process.