The sum of three terms of an Arithmetic Progression $(AP)$ is $18$ and the sum of their squares is $158$. The largest term is.......

If the $19^{th}$ term of a non-zero $A.P.$ is zero,then the ratio of its ($49^{th}$ term) to ($29^{th}$ term) is:

$A$ person is to count $4500$ currency notes. Let $a_n$ denote the number of notes he counts in the $n^{th}$ minute. If $a_1 = a_2 = \ldots = a_{10} = 150$ and $a_{10}, a_{11}, \ldots$ are in an $A.P.$ with common difference $-2$,then the time taken by him to count all notes is ............... $minutes$.

If $S_1, S_2$ and $S_3$ are the sums of the first $n_1, n_2$ and $n_3$ terms of an arithmetic progression respectively,then $\frac{S_1}{n_1}(n_2 - n_3) + \frac{S_2}{n_2}(n_3 - n_1) + \frac{S_3}{n_3}(n_1 - n_2) = ....$

If the sum of $n$ terms of an $A.P.$ is $2n^2 + 5n$,then the $n^{th}$ term will be

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: