A square is drawn by joining the midpoints of the sides of a

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A square is drawn by joining the midpoints of the sides of a [#permalink]

15 Oct 2010, 07:27

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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?

A. 18
B. 32
C. 36
D. 64
E. None

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Re: a square is drawn [#permalink]

15 Oct 2010, 08:00

B. 32

form GP s^2 + 2 (s/2)^2 + 4 (s/4)^2

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Re: a square is drawn [#permalink]

15 Oct 2010, 08:21

anilnandyala wrote:

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 indefinately . if a side of the first square is 4 cm. detemine the sum of areas of all squares?
a 18
b 32
c 36
d 64
e none

Let the side of the first square be a, so its area will be area_1=a^2;
Next square will have the diagonal equal to a, so its area will be area_2=\frac{d^2}{2}=\frac{a^2}{2};
And so on.

So the areas of the squares will form infinite geometric progression: a^2, \frac{a^2}{2}, \frac{a^2}{4}, \frac{a^2}{8}, \frac{a^2}{16}, ... with common ration equal to \frac{1}{2}.

For geometric progression with common ratio |r|<1, the sum of the progression is sum=\frac{b}{1-r}, where b is the first term.

So the sum of the areas will be sum=\frac{a^2}{1-\frac{1}{2}}=\frac{4^2}{\frac{1}{2}}=32.

Answer: B.
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Re: a square is drawn [#permalink]

17 Oct 2010, 20:17

Straight 32. The Area of square created by joining the mid points of a square is half of the bigger square.

So it becomes 16+8+4+......
Apply Infinite GP formula a/(1-r). a = 16, r = 0.5
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Re: a square is drawn [#permalink]

18 Oct 2010, 00:06

Are we supposed to know this for GMAT.

Re: a square is drawn [#permalink] 18 Oct 2010, 00:06

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A square is drawn by joining the midpoints of the sides of a

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A square is drawn by joining the midpoints of the sides of a

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