Let $a_{n}$ be the $n^{\text {th }}$ term of a G.P. of positive terms.
If $\sum\limits_{n=1}^{100} a_{2 n+1}=200$ and $\sum\limits_{n=1}^{100} a_{2 n}=100,$ then $\sum\limits_{n=1}^{200} a_{n}$ is equal to
$225$
$175$
$300$
$150$
The sum of the series $5.05 + 1.212 + 0.29088 + ...\,\infty $ is
If the sum and product of four positive consecutive terms of a $G.P.$, are $126$ and $1296$, respectively, then the sum of common ratios of all such $GPs$ is $.........$.
If the range of $f(\theta)=\frac{\sin ^4 \theta+3 \cos ^2 \theta}{\sin ^4 \theta+\cos ^2 \theta}, \theta \in \mathbb{R}$ is $[\alpha, \beta]$, then the sum of the infinite $G.P.$, whose first term is $64$ and the common ratio is $\frac{\alpha}{\beta}$, is equal to...........
Let $P(x)=1+x+x^2+x^3+x^4+x^5$. What is the remainder when $P\left(x^{12}\right)$ is divided by $P(x)$ ?
The numbers $(\sqrt 2 + 1),\;1,\;(\sqrt 2 - 1)$ will be in