Question 1191486: Each circle in an infinite sequence with decreasing radii is tangent externally to the one following it and to both sides of a given right angle. What is the ratio of the area of the first circle to the sum of the areas of the other circles in the sequence?
Found 2 solutions by math_tutor2020, greenestamps:
Answer by math_tutor2020(3816)   (Show Source):
You can put this solution on YOUR website!
Refer to this figure
The diagram is to scale.
The larger circle is of radius 1, centered at point A.
A smaller tangent circle is to the bottom left centered at B, and it has radius k.
Both circles are tangent to the x and y axis at the same time.
The circles are tangent to each other.
Points A,B,C,D are collinear.
The sub-goal is to find the radius of the smaller circle to see how it connects to the larger one.
We have these facts to consider- AE = AD = 1 = radius of the larger circle
- CE = 1 because triangle ACE is an isosceles right triangle
- FB = BD = k = radius of the smaller circle
- CF = k because triangle CFB is an isosceles right triangle
Furthermore,- AC = sqrt(2) due to the pythagorean theorem for right triangle ACE
- CB = k*sqrt(2) due to the pythagorean theorem for right triangle CFB
Let's add up the diagonal pieces of CB,BD, and DA
CB+BD+DA = k*sqrt(2) + k + 1
This is equal to the total length of AC, which we found earlier was sqrt(2)
Therefore, we can form this equation
CB+BD+DA = AC
k*sqrt(2) + k + 1 = sqrt(2)
Let's solve for k
k*sqrt(2) + k + 1 = sqrt(2)
k*(sqrt(2) + 1) + 1 = sqrt(2)
k*(sqrt(2) + 1) = sqrt(2) - 1
k = (sqrt(2) - 1)/(sqrt(2) + 1)
It's handy to rationalize the denominator
To conclude things so far, we found that if the larger circle has radius 1, then the smaller circle has radius of exactly 
If the radius of the largest circle is 1, then its area is 
Let's compute the area of the smaller circle.
Divide the two circle areas:
This is the common ratio when concerning the geometric sequence of areas of the circles.
Why? Because the common ratio involves dividing any term by its previous term.
When I computed the smaller/larger, that's exactly leading to the common ratio.
This common ratio allows us to then frame things in terms of the first circle having area of 1 square unit, the second circle having area of 1*(17-12*sqrt(2)) square unit and so on.
In other words,
First circle area = 1
second circle area = 17-12*sqrt(2)
third circle area = (17-12*sqrt(2))^2
fourth circle area = (17-12*sqrt(2))^3
etc.
A more general scenario is
area of circle N = (17-12*sqrt(2))*(area of circle N-1)
or
area of circle N+1 = (17-12*sqrt(2))*(area of circle N)
or
area of Nth circle = (17-12*sqrt(2))*(area of circle just before Nth circle)
We found that the second term, or second area, is exactly
This can be treated as the first term of the geometric sequence involving circle 2, circle 3, ...
This is because we want to find the sum of these infinitely many circles.
To sum infinitely many terms of a geometric series, we use this formula
The formula can only be used when |r| < 1 or -1 < r < 1
In this case, we have r = 17-12*sqrt(2) = 0.0294 approximately, showing that -1 < r < 1 is true and it allows us to use that formula.
So,
Summing the areas of circle 2, circle 3, circle 4, ... will have the infinite sum approach
The ratio of the first area  to the sum of the other areas  is )
This ratio is equivalent to  after multiplying both parts of the ratio by 8/pi
If you want the decimal form of this ratio, then it would be roughly  and you can divide both parts by 8 to end up with this approximation of 
The second ratio mentioned says that the area sum of the infinitely many other circles (everything but circle 1) is roughly 3.033% of the area of the first circle.
There are many ways to express the final answer, but I think the best way is to write since I think it's the most descriptive. Also, many standard ratios often involve the first value being 1. However, if your teacher requires exact values, then use one of the forms with square roots involved.
Edit: The solution greenestamps got is another way of writing the answer equivalent to the other forms I mentioned above.
Note that 16+12*sqrt(2) = 32.9705627484771 which has the reciprocal of
1/32.9705627484771 = 0.03033008588991
Answer by greenestamps(13196)  (Show Source):
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