If $x, y, z$ are three real numbers with the same sign,then the value of $\frac{x}{y} + \frac{y}{z} + \frac{z}{x}$ lies in which interval?

The common difference of the $A.P.: a_{1}, a_{2}, ..., a_{m}$ is $13$ more than the common difference of the $A.P.: b_{1}, b_{2}, ..., b_{n}$. If $b_{31} = -277$, $b_{43} = -385$ and $a_{78} = 327$, then $a_{1}$ is equal to

The sum of all two-digit positive numbers which,when divided by $7$,yield $2$ or $5$ as a remainder is:

Let $x, y > 0$. If $x^{3} y^{2} = 2^{15}$,then the least value of $3x + 2y$ is

In an $A.P.$,if the $p^{\text{th}}$ term is $\frac{1}{q}$ and the $q^{\text{th}}$ term is $\frac{1}{p}$,prove that the sum of the first $pq$ terms is $\frac{1}{2}(pq+1)$,where $p \neq q$.

The sum of $1 + 3 + 5 + 7 + \dots$ up to $n$ terms is