Question

The quotient when $1+x^2+x^4+x^6+....+x^{34}$ is divided by $1+x+x^2+x^3+....+x^{17}$ is

• A. $x^{17}-x^{15}+x^{13}-x^{11}+.....+x$
• B. $x^{17}+x^{15}+x^{13}+x^{11}+.....+x$
• C. $x^{17}+x^{16}+x^{15}+x^{14}+.....+x$
• D. $x^{17}-x^{16}+x^{15}-x^{14}+.....+x$

Answer 1

Answer: (D)

$x^{17}-x^{16}+x^{15}-x^{14}+.....+x$

$1,x^2,x^4,..........,x^{34}$ forms a finite geometric series with common ratio $x^2$

Now, sum of finite geometric series $1,r,r^2,.....,r^{n-1}$ with common ration $r$ is given by

$1+r+r^2+.....+r^{n-1}=\frac{1-r^n}{1-r}$ ...... $\left(1\right)$

Then, we have

$1+x^2+x^4+\dots x^{34}=\frac{(x^2{)}^{18}-1}{x^2-1}=\frac{x^{36}-1}{x^2-1}\dots \dots \left(2\right)$

and

$1+x+x^2+\dots x^{17}=\frac{x^{18}-1}{x-1}\dots \dots \left(3\right)$

Dividing $\left(2\right)$ with $\left(3\right)$, we get

$\Rightarrow \frac{1+x^{18}}{1+x}=x^{17}-x^{16}+........+x$