8. sequence and series — MATHS STD 11 Science — Question

Let,$P(x)=1+x^2+x^4+x^6+\ldots+x^{22}$

$Q(x)=1+x+x^2+x^3+\ldots+x^{11}$

$P(x)=\frac{x^{24}-1}{x^2-1}$

$\left[\because a+a r+a r^2+\ldots+a r^{n-1}=\frac{a\left(r^n-1\right)}{r-1}\right]$

Similarly, $Q(x)=\frac{x^{12}-1}{x-1}$

$\therefore \quad \frac{P(x)}{Q(x)}=\left(\frac{x^{24}-1}{x^2-1}\right)\left(\frac{x-1}{x^{12}-1}\right)=\frac{x^{12}+1}{x+1}$

Now, $\left(1+x^{12}\right)$ is divided by $(x+1)$, then by remainder theorem remainder is $2$

Now, $P(x)$ is divided by $Q(x)$, then

remainder $=2\left(1+x^2\right)\left(1+x^4+x^8\right)$

$=2\left(1+x^2+x^4+\ldots+x^{10}\right)$