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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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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 $$sum=\frac{a^2}{1-\frac{1}{2}}=\frac{4^2}{\frac{1}{2}}=32$$.

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B. 32

form GP s^2 + 2 (s/2)^2 + 4 (s/4)^2
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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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Bumping for review and further discussion*. Get a kudos point for an alternative solution!

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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.
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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?
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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 96480 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 $$\frac{diagonal^2}{2}=\frac{a^2}{2}$$.

Hope it's clear.
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Re: A square is drawn by joining the midpoints of the sides of a  [#permalink]

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area of the first square is 4^2 = 16.
the second one, has the sides 2*sqrt(2), and area 8.
so area is >24.
we can eliminate right away A and E.
the third square will have a side of 2, and area 4.
24+4 = 28.
fourth one will have sides 1, and area 1.
28+1=29.
now, here get's more interesting, since the pattern continues infinitely, and the areas of the squares get decreased at an incredible rate, it must be the closest # to be the sum of areas. the closest number to 29 is 32.
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Re: A square is drawn by joining the midpoints of the sides of a  [#permalink]

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Bunuel wrote:
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 $$sum=\frac{a^2}{1-\frac{1}{2}}=\frac{4^2}{\frac{1}{2}}=32$$.

Dear Bunuel, Do you have a thread that says it all about geometric progressions? If yes, could you link it to me, please?
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Posts: 58335
Re: A square is drawn by joining the midpoints of the sides of a  [#permalink]

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paolodeppa wrote:
Bunuel wrote:
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 $$sum=\frac{a^2}{1-\frac{1}{2}}=\frac{4^2}{\frac{1}{2}}=32$$.

Dear Bunuel, Do you have a thread that says it all about geometric progressions? If yes, could you link it to me, please?

12. Sequences

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

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Sides of subsequent squares:
4, 2√2, 2, √2, 1, and so on

Areas: 16, 8, 4, 2, 1, 0.5, and so on

GP series with
first term a1= 16
Common ratio r= 1/2
No of terms n= infinity

Sum of terms= a1(r^n-1)/(r-1)

Now, r^n = (1/2)^infinity will tends to zero as denominator 2 tends to infinity.

So, Sum= a1(0-1)/(-0.5) = 16*2= 32.

Ans B

Posted from my mobile device Re: A square is drawn by joining the midpoints of the sides of a   [#permalink] 08 Feb 2019, 13:53
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