If $a_{1} (>0), a_{2}, a_{3}, a_{4}, a_{5}$ are in a $G$.$P$.,$a_{2} + a_{4} = 2a_{3} + 1$ and $3a_{2} + a_{3} = 2a_{4}$,then $a_{2} + a_{4} + 2a_{5}$ is equal to

Let $\alpha, \beta, \gamma$ $(\alpha < \beta < \gamma)$ be roots of $ax^3+bx^2+cx+d=0$ and $u, v, w$ $(u < v < w)$ be roots of $ak^3x^3+bk^2x^2+ckx+d=0$. If $\beta^2=\alpha \gamma$,then:

If three successive terms of a $G.P.$ with common ratio $r$ $(r > 1)$ are the lengths of the sides of a triangle,and $[r]$ denotes the greatest integer less than or equal to $r$,then $3[r] + [-r]$ is equal to:

Let $A_{1}, A_{2}, A_{3}, \ldots$ be squares such that for each $n \geq 1,$ the length of the side of $A_{n}$ equals the length of the diagonal of $A_{n+1}$. If the side length of $A_{1}$ is $12 \text{ cm}$,then the smallest value of $n$ for which the area of $A_{n}$ is less than $1 \text{ cm}^2$ is:

Let $A_n = \left( \frac{3}{4} \right) - \left( \frac{3}{4} \right)^2 + \left( \frac{3}{4} \right)^3 - \dots + (-1)^{n-1} \left( \frac{3}{4} \right)^n$ and $B_n = 1 - A_n$. Then,the least odd natural number $p$ such that $B_n > A_n$ for all $n \geq p$ is:

If the sum of an infinite $G.P.$ and the sum of the squares of its terms is $3$,then the common ratio of the first series is