If $\frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \ldots$ up to $\infty = \frac{\pi^2}{6}$, then the value of $\frac{1}{1^2} + \frac{1}{3^2} + \frac{1}{5^2} + \ldots$ up to $\infty$ is

It is given that the sequence {$x_n$} satisfies $x_1 = 0, x_{n+1} = x_n + 1 + 2√(1+x_n)$ for $n = 1,2,...$ Then $x_{31}$ is _______

The terms of a geometric progression are real and positive. If the $p$-th term of the progression is $q$ and the $q$-th term is $p$, then the logarithm of the first term is

The numbers $-16,2^{x+3}-2^{2 x-1}-16,2^{2 x-1}+16$ are in an arithmetic progression. Then $x$ equals ________.

Let $\alpha, \beta$ be the roots of $x^2 - x + p = 0$ and $\gamma, \delta$ be the roots of $x^2 - 4x + q = 0$ where p and q are integers. If $\alpha, \beta, \gamma, \delta$ are in geometric progression then $p + q$ is

The sum of the first 5 terms of a geometric progression is the same as the sum of the first 7 terms of the same progression. If the sum of the first 9 terms is 24, then the 4th term of the progression is

Let $S$ denote an arithmetic progression whose first term is either 132 or 158, and the common difference is an even integer less than 10. If the $n^{\text{th}}$ term of $S$ is 174, then the number of possible distinct values of $n$ is ___