The famous mathematician associated with finding the sum of the first $100$ natural numbers is

Find the sum of the first $20$ terms of the $A.P.$ $50, 46, 42, \ldots$

For the $A.P.$ $-2, -4, -6, -8, \ldots,$ the common difference is............

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 $0$.

If the $n$th terms of the two $APs$: $9, 7, 5, \ldots$ and $24, 21, 18, \ldots$ are the same,find the value of $n$. Also,find that term.

The sum of the first $n$ terms of an $AP$ whose first term is $8$ and the common difference is $20$ is equal to the sum of first $2n$ terms of another $AP$ whose first term is $-30$ and the common difference is $8$. Find $n$.