If $x+1, 3x$ and $4x+2$ are three consecutive terms of an $A.P.$,find the value of $x$.

The sum of the first $20$ terms of the $A.P.$ $5, 11, 17, 23, \ldots$ is:

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 $A.P.$

Verify that each of the following is an $AP$,and then write its next three terms.
$a+b, (a+1)+b, (a+1)+(b+1), \ldots$

Two $AP$s have the same common difference. The first term of one of these is $-1$ and that of the other is $-8$. Then the difference between their $4^{\text{th}}$ terms is

If the $2^{\text{nd}}$ term of an $AP$ is $13$ and the $5^{\text{th}}$ term is $25$,what is its $7^{\text{th}}$ term?