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# Test of Mathematics Solution Subjective 170 - Infinite Circles  This is a Test of Mathematics Solution Subjective 170 (from ISI Entrance). The book, Test of Mathematics at 10+2 Level is Published by East West Press. This problem book is indispensable for the preparation of I.S.I. B.Stat and B.Math Entrance.

Also visit: I.S.I. & C.M.I. Entrance Course of Cheenta

## Problem

Let be an infinite sequence of circles lying in the positive quadrant of the -plane, with strictly decreasing radii and satisfying the following conditions. Each touches both the -axis and the -axis. Further, for all , the circle touches the circle externally. If has radius , then show that the sum of the areas of all these circles is .

## Solution

Consider the following diagram where the Green line segment is , the radius of the circle, and the Yellow line segment is . As we are told about the symmetricity of the figure in the problem we can say that:   Let's say .

Now the total sum of the areas of the circles is: Now as , we can say that: as .

Substituting the value of and we have,

Sum = .

Hence Proved. This is a Test of Mathematics Solution Subjective 170 (from ISI Entrance). The book, Test of Mathematics at 10+2 Level is Published by East West Press. This problem book is indispensable for the preparation of I.S.I. B.Stat and B.Math Entrance.

Also visit: I.S.I. & C.M.I. Entrance Course of Cheenta

## Problem

Let be an infinite sequence of circles lying in the positive quadrant of the -plane, with strictly decreasing radii and satisfying the following conditions. Each touches both the -axis and the -axis. Further, for all , the circle touches the circle externally. If has radius , then show that the sum of the areas of all these circles is .

## Solution

Consider the following diagram where the Green line segment is , the radius of the circle, and the Yellow line segment is . As we are told about the symmetricity of the figure in the problem we can say that:   Let's say .

Now the total sum of the areas of the circles is: Now as , we can say that: as .

Substituting the value of and we have,

Sum = .

Hence Proved.

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