math gre question

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in an infinite series of circles the radius of the second circle is one half the radius of the first circle and the radius of the third circle is one half the radius of the second circle. if the first circle has a radius of one inch which of the following statements best describe the sum of the areas of all the circles?

1. the sum of the area of all the circle converge to pi in^2

2. the sum of the areas of all the circles diverge from 2pi in^2

3. the sum of the areas of all the circles converge to 2pi in^2

4. the sum of the areas of all the circles diverge from pi in^2

my solution: i think it is number 4 because the radius ends up being zero.

• math gre question -

a total of 920 adult and student ticket were sold for the school play. the number of student tickets sold was 16 less than twice the number of adult tickets.how many adult tickets were sold?

• math gre question -

sum is pi(1^2 + (1/2)^2 + (1/4)^2 + ...)
= pi (1 + 1/4 + 1/16 + ...)
= pi (1/(1 - 1/4)) = pi (1/(3/4)) = 4pi/3

I think the answer they wanted was 2pi (#3) but 2 = 1 + 1/2 + 1/4 + 1/8 + 1/16 + ...
and we have 1 + 1/4 + 1/16 + 1/64 + ...

a + s = 920
s = 2a-16
a + 2a-16 = 920
3a = 936
a = 312
s = 308 = 2*312 - 16

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