Nusret Balcı TMUA Paper 1 · Question 2

A sequence of similar sectors $S_1,S_2,S_3,\ldots$ has a constant central angle $\theta$ (in radians). The radius $r_{n+1}$ of each sector is defined by the arc length $s_n$ of the previous sector.

Given that the sum of the perimeters of all sectors is exactly three times the perimeter of the first sector:

\sum_{n=1}^\infty\operatorname{Perimeter}(S_n)=3\times\operatorname{Perimeter}(S_1)

Find the ratio of the total area of all sectors to the area of the first sector:

\text{Ratio}=\frac{\sum_{n=1}^\infty\operatorname{Area}(S_n)}{\operatorname{Area}(S_1)}

2.6

3

1.1

1.9

1.8

4.3

\sqrt2

\sqrt3

examday

Nusret Balcı TMUA Paper 1 · Q2