In an $A.P.$,if the $m^{\text{th}}$ term is $n$ and the $n^{\text{th}}$ term is $m$,where $m \neq n$,find the $p^{\text{th}}$ term.

If the sum of a certain number of terms of the $A.P.$ $25, 22, 19, \ldots$ is $116$,find the last term.

If $\log_{3} 2, \log_{3} (2^{x} - 5)$ and $\log_{3} (2^{x} - \frac{7}{2})$ are in an Arithmetic Progression $(AP)$,then $x = \dots$

Let $a_{1}, a_{2}, a_{3}, \ldots$ be an $A.P.$ If $\frac{a_{1}+a_{2}+\ldots+a_{10}}{a_{1}+a_{2}+\ldots+a_{p}}=\frac{100}{p^{2}}, p \neq 10$,then $\frac{a_{11}}{a_{10}}$ is equal to :

Let $S = \{(a, b, c) \in \mathbb{N} \times \mathbb{N} \times \mathbb{N} : a+b+c=21, a \leq b \leq c\}$ and $T = \{(a, b, c) \in \mathbb{N} \times \mathbb{N} \times \mathbb{N} : a, b, c \text{ are in } AP\}$, where $\mathbb{N}$ is the set of all natural numbers. Then, the number of elements in the set $S \cap T$ is:

If the sum of the first $n$ terms of a sequence is of the form $An^2 + Bn$,where $A$ and $B$ are constants independent of $n$,then the sequence is a ........