The $20^{th}$ term of the $A.P.$ $1, 11, 21, \ldots$ is........

From natural numbers $1$ to $288,$ find the number $x$ such that the sum of all natural numbers smaller than $x$ is equal to the sum of all natural numbers greater than $x$ but less than or equal to $288.$

How many terms of the $AP: -15, -13, -11, \ldots$ are needed to make the sum $-55$? Explain the reason for the double answer.

The sequence $1, 3, 6, 10, \ldots$ is

What is the common difference of an $AP$ in which $a_{18}-a_{14}=32$?

In an $A.P.$,ten times the $10^{th}$ term equals fifteen times the $15^{th}$ term. Prove that the $25^{th}$ term of the $A.P.$ is $0$.