# Consider a series of n concentric circles C(1),C(2),C(3), ....., C(n) with radii r(1),r(2),r(3), ......., r(n) respectively, satisfying r(1)>r(2)>r(3).... > r(n) & r(1)=10. The circles are such that the chord of contact of tangents from any point on Ci to C(i+1) is a tangent to C(i+2) (i=1,2,3,...). Find the value of r(1)+r(2)+r(3)+.........r(infinity), if the angle between the tangents from any point of C(1) to C(2) is 60.

Arun Kumar IIT Delhi
$\\ \\consider making the diagram yourself \\you must know that c(n) is the innermost circle \\i'm assuming tangents form c(i) to c(i+1) of which chord \\of contact touches c(i+2) \\try to understand the analysis \\consider the triangle made by r(i+1),r(i) and length of tangent \\angle between length of tangent and r(i)=30 \\(since 60^0 is given) \\so sin30={r(i+1) \over r(i)} \\r(i+1)={r(i) \over 2} \\consider one more triangle by r(i+1),half of chord and \\r(i+2) \\again cos(60)={r(i+2) \over r(i+1)} \\=>r(i+2)={r(i+1) \over 2} \\=>GP with common ratio 1/2 \\=>sum={a \over 1-r}={10 \over (1-1/2)}=20 \\$