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
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
.
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
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
.
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.