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

Do you really need a hint? Try it first!

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.

Similar Problem

cos(sin(x)) function in ISI Entrance

A simple trigonometric equation from ISI Entrance. Try this problem. We also added a quiz, some related problems, and finally video.

Geometry of AM GM Inequality

AM GM Inequality has a geometric interpretation. Watch the video discussion on it and try some hint problems to sharpen your skills.

Counting triangles in ISI Entrance

Can you combine geometry and combinatorics? This ISI Entrance problems requires just that. We provide sequential hints, additional problems and video.

Paper folding geometry in ISI Entrance

A problem from ISI Entrance that requires Paper folding geometry. We provide sequential hints so that you can try the problem!

An Hour of Beautiful Proofs

Every week we dedicate an hour to Beautiful Mathematics - the Mathematics that shows us how Beautiful is our Intellect. This week, I decided to do three beautiful proofs in this one-hour session... Proof of Fermat's Little Theorem ( via Combinatorics )It uses...

Inequality – In Equality

This article aims to give you a brief overview of Inequality, which can be served as an introduction to this beautiful sub-topic of Algebra. This article doesn't aim to give a list of formulas and methodologies stuffed in single baggage, rather it is specifically...

AM GM inequality in ISI Entrance

Arithmetic Mean and Geometric Mean inequality form a foundational principle. This problem from I.S.I. Entrance is an application of that.

How to solve an Olympiad Problem (Number Theory)?

Suppose you are given a Number Theory Olympiad Problem. You have no idea how to proceed. Totally stuck! What to do? This post will help you to atleast start with something. You have something to proceed. But as we share in our classes, how to proceed towards any...

How are Bezout’s Theorem and Inverse related? – Number Theory

The inverse of a number (modulo some specific integer) is inherently related to GCD (Greatest Common Divisor). Euclidean Algorithm and Bezout’s Theorem forms the bridge between these ideas. We explore these beautiful ideas.

How to use Invariance in Combinatorics – ISI Entrance Problem

Invariance is a fundamental phenomenon in mathematics. In this combinatorics problem from ISI Entrance, we discuss how to use invariance.