The value of $0.\mathop {234}\limits^{\,\,\, \bullet \,\, \bullet } $ is
$\frac{{232}}{{990}}$
$\frac{{232}}{{9990}}$
$\frac{{232}}{{900}}$
$\frac{{232}}{{9909}}$
If $a _{1}(>0), a _{2}, a _{3}, a _{4}, a _{5}$ are in a G.P., $a _{2}+ a _{4}=2 a _{3}+1$ and $3 a _{2}+ a _{3}=2 a _{4}$, then $a _{2}+ a _{4}+2 a _{5}$ is equal to
If the first term of a $G.P. a_1, a_2, a_3......$ is unity such that $4a_2 + 5a_3$ is least, then the common ratio of $G.P.$ is
${7^{th}}$ term of the sequence $\sqrt 2 ,\;\sqrt {10} ,\;5\sqrt 2 ,\;.......$ is
The ${4^{th}}$ term of a $G.P.$ is square of its second term, and the first term is $-3$ Determine its $7^{\text {th }}$ term.
Let $S_1$ be the sum of areas of the squares whose sides are parallel to coordinate axes. Let $S_2$ be the sum of areas of the slanted squares as shown in the figure. Then, $\frac{S_1}{S_2}$ is equal to