For the $A.P.$ $3, 8, 13, 18, \ldots$,$T_{n} = \ldots$

For an $A.P.$,five times the $5^{th}$ term is equal to eight times the $8^{th}$ term. Find the $13^{th}$ term of the $A.P.$

For a given finite $A.P.$,$a=1, l=10$ and $n=10$. Then,$S_{10} = ........$

Two APs have the same common difference. The first term of one $AP$ is $2$ and that of the other is $7$. The difference between their $10^{\text{th}}$ terms is the same as the difference between their $21^{\text{st}}$ terms,which is the same as the difference between any two corresponding terms. Why?

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 $S_{3n} = 3(S_{2n} - S_n)$.

The $20^{th}$ term of the $A.P.$ $2, -2, -6, -10, \ldots$ is.......