Which term of the $AP: 21, 42, 63, 84, \ldots$ is $210$?

Justify whether it is true to say that $-1, -\frac{3}{2}, -2, \frac{5}{2}, \ldots$ forms an $AP$ as $a_{2}-a_{1} = a_{3}-a_{2}$.

In an $AP$,if $a = 1$,$a_{n} = 20$,and $S_{n} = 399$,then $n$ is:

Let $S_n$,$S_{2n}$,and $S_{3n}$ be the sums of $n$,$2n$,and $3n$ terms of an Arithmetic Progression $(AP)$,respectively. Prove that $S_{3n} = 3(S_{2n} - S_n)$.

Solve the equation $-4 + (-1) + 2 + \ldots + x = 437$.

Find five numbers in $A.P.$ such that their sum is $30$ and the sum of their squares is $220$.