7.5 Infinite geometric series (in class)

Consider a geometric sequence [math]\{a_n\}[/math] that satisfies [math]a_1+a_2+a_3=18[/math] and [math]a_2+a_3+a_4=-9[/math]. Find the infinite series of [math]\{a_n\}[/math].

The value of [math]\dfrac{1}{11}+\dfrac{1}{101}+\dfrac{1}{1001}+\dfrac{1}{10001}+\cdots[/math] is

Observe the figure above. Assume that the greatest circle is a unit circle. Find the summation of the area of all circles.