Let $a_1, a_2, a_3, \dots$ be an $A$.$P$. and $g_1, g_2, g_3, \dots$ be an increasing $G$.$P$. If $a_1 = g_1$ and $a_2 + g_2 = 1$ and $a_3 + g_3 = 4$, then $a_{10} + g_5$ is equal to:

The sum of all $3$-digit numbers less than or equal to $500$,formed without using the digit $1$,which are also multiples of $11$,is ..... .

The number of common terms in the progressions $4, 9, 14, 19, \ldots$ up to $25^{\text{th}}$ term and $3, 6, 9, 12, \ldots$ up to $37^{\text{th}}$ term is:

Consider the sequence $a_{1}, a_{2}, a_{3}, \ldots$ such that $a_{1}=1, a_{2}=2$ and $a_{n+2}=\frac{2}{a_{n+1}}+a_{n}$ for $n=1, 2, 3, \ldots$. If $\left(\frac{a_{1}+\frac{1}{a_{2}}}{a_{3}}\right) \cdot\left(\frac{a_{2}+\frac{1}{a_{3}}}{a_{4}}\right) \cdot\left(\frac{a_{3}+\frac{1}{a_{4}}}{a_{5}}\right) \cdots\left(\frac{a_{30}+\frac{1}{a_{31}}}{a_{32}}\right)=2^{\alpha}\left({}^{61}C_{31}\right)$,then $\alpha$ is equal to.

Let $a_{1}=1$ and for $n \ge 1$, $a_{n+1} = \frac{1}{2}a_{n} + \frac{n^{2}-2n-1}{n^{2}(n+1)^{2}}$. Then $|\sum_{n=1}^{\infty}(a_{n}-\frac{2}{n^{2}})|$ is equal to ........... .

If $x = \sum_{n=0}^{\infty} (-1)^{n} \tan^{2n} \theta$ and $y = \sum_{n=0}^{\infty} \cos^{2n} \theta$ for $0 < \theta < \frac{\pi}{4}$,then: