If the sum of the first four terms of an $A.P.$ is $6$ and the sum of its first six terms is $4$, then the sum of its first twelve terms is

Let $A = \{1, a_{1}, a_{2}, \ldots, a_{18}, 77\}$ be a set of integers with $1 < a_{1} < a_{2} < \ldots < a_{18} < 77$. Let the set $A + A = \{x + y : x, y \in A\}$ contain exactly $39$ elements. Then,the value of $a_{1} + a_{2} + \ldots + a_{18}$ is equal to:

The sides of a right-angled triangle are in arithmetic progression. If the triangle has an area of $24$,then what is the length of its smallest side?

The sum of the first $20$ terms common between the series $3 + 7 + 11 + 15 + \dots$ and $1 + 6 + 11 + 16 + \dots$ is

What is the sum of $n$ arithmetic means between $a$ and $b$?

The sum of the first four terms of an $A.P.$ is $56$. The sum of the last four terms is $112$. If its first term is $11$,then find the number of terms.