Given that $1 + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + ... = \frac{\pi^2}{6}$, the value of $1 + \frac{1}{3^2} + \frac{1}{5^2} + \frac{1}{7^2} + ...$ is

Let $S_n$ be sum of the first $n$ terms of an A.P. If $S_5 = S_9$, what is the ratio of $a_3 : a_5$

The number of terms common to both the arithmetic progressions $2, 5, 8, 11, ..., 179$ and $3, 5, 7, 9, ..., 101$ is

Assume that all positive integers are written down consecutively from left to right as in 1234567891011...... The 6389th digit in this sequence is

The sum of a given infinite geometric progression is 80 and the sum of its first two terms is 35. Then the value of $n$ for which the sum of its first $n$ terms is closest to 100, 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

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