Question: A square is drawn by joining the midpoints of the sides of a given square. A third square is drawn inside the second square in the way and this process is continued indefinitely. If a side of the first square is 4 cm. determine the sum of areas of all squares?

1. 18
2. 32
3. 36
4. 64
5. None

Correct Answer: (B)
Solution with Explanation:
Approach Solution : 1

The Sides of subsequent squares are 4, 2√2, 2, √2, 1, and so on.
The areas are 16, 8, 4, 2, 1, 0.5, and so on.
The GP series with the first term a1= 16
The common ratio r= ½
Number of terms n= infinity
Now, the sum of terms= a1(r^n – 1)/(r – 1)
Therefore r^n = (1/2)^infinity tending to zero like denominator 2 tending to infinity
As a result, the sum of area of all squares = [a1(0 – 1)] / (-0.5) = 16 * 2 = 32.

Approach Solution : 2

Consider the side of the first square be a, so the area of the first square is a^2
The following square will have the diagonal equal to a, so the area of this will be (d^2)/2 = (a^2)/2
And so it will continue.
As a result, an infinite geometric progression will be formed from the areas of the squares.
Now we have a^2, (a^2)/2, (a^2)/4, (a^2)/8, (a^2)/16.... and the common ration will be ½
For the geometric progression that has the common ratio |r| < 1|,
the sum of the progression is b / (1−r). Note that here b is the first term.
As a result, the sum of the areas of the squares = [a^2] / [1-(1/2)] = (4^2) / (1/2) = 32

Approach Solution : 3

The area of the first square is 4^2 = 16.
The area of the second square is (2√2) ^2 = 8.
Therefore the sum of areas is greater than 18
So the first and last options are eliminated
The third square will have a side of 2, and so the area = 4
Now the sum of the areas of the squares till now is, 24 + 4 = 28
The fourth square will have sides 1, and so the area = 1
Now the sum of the areas of the squares till now is, 28 + 1=29.
It must be the closest number to be the sum of areas because the pattern repeats indefinitely and the areas of the squares shrink at an astounding rate.
32 is the closest number to 29 out of all options.

“A Square Is Drawn By Joining The Midpoints Of The Sides Of A Given Square” is a topic that is covered in the quantitative reasoning section of the GMAT. To successfully execute the GMAT Problem Solving questions, a student must possess a wide range of qualitative skills. The entire GMAT Quant section consists of 31 questions. The problem-solving section of the GMAT Quant topics requires the solution of calculative mathematical problems.

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