# Understand the problem

For each natural number k, choose a complex number \(z_k\) , with\( | z_k | = 1 \), and denote \( a_k \) by the area of the triangle formed by \(z_k , i \cdot z_k , z_k + i \cdot z_k \) . Then which of the following is true for the series below: $$ \Sigma (a_k)^k $$

(1) It converges only if every \(z_k\) lies in the same quadrant

(2) It always diverges

(3) It always converges

(4) None of the above

##### Source of the problem

##### Topic

Complex Numbers and Geometry

##### Difficulty Level

6 out of 10

##### Suggested Book

Complex Numbers from A to Z by Titu Andreescu

# Start with hints

# Connected Program at Cheenta

# I.S.I. & C.M.I. Entrance Program

Indian Statistical Institute and Chennai Mathematical Institute offer challenging bachelor’s program for gifted students. These courses are B.Stat and B.Math program in I.S.I., B.Sc. Math in C.M.I.

The entrances to these programs are far more challenging than usual engineering entrances. Cheenta offers an intense, problem-driven program for these two entrances.

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