Question

A square is drawn by joining mid-points of the sides of a square. Another 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 16 cm, then what is the sum of the areas of all the squares?
1. 256 sq. cm
2. 512 sq. cm
3. 1024 sq. cm
4. 512/3 sq. cm

Answer 1

Correct Answer - Option 2 : 512 sq. cm

Concept:

A square is drawn by joining mid-points of the sides of a square. Another square is drawn inside the second square in the same way and the process is continued indefinitely. 

The side of the first square = a 

Side of the second square = \(\rm \dfrac a {\sqrt2}\) and so on....

If the series is in GP with common ratio r and first term a then sum of infinity GP = \(\rm \dfrac a {1-r}\)

Calculations:

Given, a square is drawn by joining mid-points of the sides of a square. Another square is drawn inside the second square in the same way and the process is continued infinity.

The side of the first square is 16 cm.

Side of second square = \(\rm \dfrac {16}{\sqrt2}\) = 8√2 cm.

Side of third square = 8 cm. and so on... 

Sum of area of squares = (16)² + (8 √2)² + (8)² + ....

Sum of area of squares = 256 + 128 + 64 + ....

The series is in GP

⇒Sum of the area of squares = \(\dfrac {256}{1-\dfrac12}\)

⇒Sum of the area of squares = 512