   Chapter 11.2, Problem 79E

Chapter
Section
Textbook Problem

The figure shows two circles C and D of radius 1 that touch at P. The line T is a common tangent line; C 1 is the circle that touches C, D, and T; C 2 is the circle that touches C, D, and C 1 ; C 3 is the circle that touches C, D, and C 2 . This procedure can be continued indefinitely and produces an infinite sequence of circles { C n } . Find an expression for the diameter of C n and thus provide another geometric demonstration of Example 8. To determine

To find:

An expression for the diameter of Cn

Explanation

1) Concept:

Use the Pythagorean Theorem and the induction method

If sequence Sn is convergent and limnSn=s  exists as a real number, then the series n=1an is convergent and n=1an=s

2) Given:

3) Calculation:

Let dn be the diameter of Cn

Draw the line from the center of  Ci to the center of D as shown below:

By using the Pythagorean Theorem,

12+1-12d12=1+12d12

1=1+12d12-1-12d12

1=1+d1+d124-1+d1-d124

1=2d1

d1=12

Similarly, for C2

12+1-d1-12d22=1+12d22

1=1+12d22-1-d1-12d22

1=2d2+2d1-d12-d1d2

1=2-d1d1+d2

Multiply both sides by 2-d1, it becomes

12-d1=d1+d2

Subtract d1 from both sides,

d2=12-d1-d1

d2=1-d122-d1

Substitute d1=12 in the above step

d2=1-1222-12

d2=16=12·3

Similarly, for C3

12+1-d1-d2-12d3<

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