The sides of a triangle are distinct positive integers in an arithmetic progression. If the smallest side is $10$,the number of such triangles is

The $r$-th term of an arithmetic progression is $T_r$. Its first term is $a$ and the common difference is $d$. If for some positive integers $m, n, m \neq n,$ we have $T_m = 1/n$ and $T_n = 1/m,$ then $a - d = \dots\dots.$

The difference between any two consecutive interior angles of a polygon is $5^{\circ}$. If the smallest angle is $120^{\circ}$,find the number of the sides of the polygon.

Let $a_1, a_2, a_3, \ldots$ be in an arithmetic progression of positive terms. Let $A_{k}=a_1^2-a_2^2+a_3^2-a_4^2+\ldots+a_{2k-1}^2-a_{2k}^2$. If $A_3=-153$,$A_5=-435$ and $a_1^2+a_2^2+a_3^2=66$,then $a_{17}-A_7$ is equal to:

If the $p^{th}$,$q^{th}$,and $r^{th}$ terms of an Arithmetic Progression are $a$,$b$,and $c$ respectively,then $[a(q - r) + b(r - p) + c(p - q)] = ?$

Three numbers are in $A.P.$ such that their sum is $18$ and the sum of their squares is $158$. The greatest number among them is