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Algebra Level 1

If $w \not= 1$ is an $n$ -th root of unity, then find the value of

$1+w+w^2+w^3+\cdots +w^{n-1}$

2 1 0

n

1 solution

Rishabh Jain
Jun 9, 2016

$\Large\color{#007fff}{\mathfrak{Method~1}}$ $\large{\begin{aligned}&w^n-1=0\\&\\\implies&\underbrace{(w-1)}_{\large \neq 0}(1+w+w^2+\cdots +w^{n-1})=0\\\implies&\color{#20A900}{ 1+w+w^2+\cdots +w^{n-1}=0}\end{aligned}}$

$\Large\color{#007fff}{\mathfrak{Method~2}}$ Use formula of sum of GP which gives:-

$\dfrac{w^n-1}{w-1}=0~~~~~(\small{\because w^n=1})$

Nice solution. I used method 1. Btw, there is a little typo in the last line.

展豪張 - 5 years ago

Lol ! ... Thanks .. :-)

Rishabh Jain - 5 years ago

How did you add the calligraphic font, looks great!

Mahdi Raza - 1 year, 1 month ago

Let's get United!

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