Question

Identify the remainder when $1+x+x^2+x^3+....+x^{2012}$ is divided by $x-1$

• A. 2012
• B. 1
• C. 2013
• D. 0

Answer 1

The correct option is C

2013

Let $f\left(x\right)=1+x+x^2+x^3+.....+x^{2012}$
Here, we will apply the remainder theorem.
When f(x) is divided by (x-1), the remainder will be given by f(1).
$f\left(1\right)=1+1+1^2+1^3+...+1^{2012}$
= 2013 times 1
= 2013
$\therefore$ Remainder = 2013