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A square is drawn by joining the midpoints of the sides of a [#permalink]
15 Oct 2010, 06:27

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67% (02:32) correct
33% (01:37) wrong based on 245 sessions

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?

Re: a square is drawn [#permalink]
15 Oct 2010, 07:21

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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.

Re: A square is drawn by joining the midpoints of the sides of a [#permalink]
01 Dec 2013, 01:59

Buneul, I did it as below. Firstly, construct the series: 16, 8, 4, 2, 1, 1/2(.5), 1/4(.25), 1/8(.125), 1/16(.06), 1/32(.03), 1/64(.01), 1/128(.00) and so on Sum from 16 to 1/64 is 31.975. Further you observe that as we move forward in the series the value goes on becoming insignificant and adds infinitesimally small values to 31.975. So, answer will also be very close to 31.975. So, 32 is the answer. However, one of the drawbacks of the method I suggested is that one has to be very good in dealing with fractions.

Re: A square is drawn by joining the midpoints of the sides of a [#permalink]
12 Dec 2013, 05:36

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WholeLottaLove wrote:

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.

I don't quite follow. You say the diagonal is equal to a (the exterior length of the largest triangle) but isn't it equal to 1/2 the diagonal * √2?

The area of a square is side^2 or \frac{diagonal^2}{2}.

Attachment:

Squares.png [ 2.4 KiB | Viewed 2411 times ]

The length of a diagonal of blue square is equal to the length of a side of black square. The area of black square is side^2=a^2 and the area of blue square is \frac{diagonal^2}{2}=\frac{a^2}{2}.

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