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

If the sum of the $3^{\text{rd}}$ and the $8^{\text{th}}$ terms of an $AP$ is $7$ and the sum of the $7^{\text{th}}$ and the $14^{\text{th}}$ terms is $-3,$ find the $10^{\text{th}}$ term.

With respect to the usual notations of an $A.P.$,if $T_{n} = 28$,$S_{n} = 144$,and $n = 9$,find the first term $a$.

Jaspal Singh repays his total loan of $Rs. 118000$ by paying every month starting with the first instalment of $Rs. 1000$. If he increases the instalment by $Rs. 100$ every month,what amount will be paid by him in the $30^{\text{th}}$ instalment? What amount of loan does he still have to pay after the $30^{\text{th}}$ instalment?

The sums of $n$ terms,$2n$ terms,and $3n$ terms of an $A.P.$ are $S_1, S_2,$ and $S_3$ respectively. Prove that $S_3 = 3(S_2 - S_1)$.

For a given $A.P.$,the first term is $-4$ and the common difference is $-5$. Then,the $12^{th}$ term of the $A.P.$ is $\ldots \ldots \ldots$.