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

Let the digits $a, b, c$ be in $A.P.$ Nine-digit numbers are to be formed using each of these three digits thrice such that three consecutive digits are in $A.P.$ at least once. How many such numbers can be formed?

Let $l_1, l_2, \ldots, l_{100}$ be consecutive terms of an arithmetic progression with common difference $d_1$, and let $w_1, w_2, \ldots, w_{100}$ be consecutive terms of another arithmetic progression with common difference $d_2$, where $d_1 d_2=10$. For each $i=1,2, \ldots, 100$, let $R_i$ be a rectangle with length $l_i$, width $w_i$ and area $A_i$. If $A_{51}-A_{50}=1000$, then the value of $A_{100}-A_{90}$ is. . . . . 

If ${\log _3}2,\;{\log _3}({2^x} – 5)$ and ${\log _3}\left( {{2^x} – \frac{7}{2}} \right)$ are in $A.P.$, then $x$ is equal to

If $p$ times the ${p^{th}}$ term of an $A.P.$ is equal to $q$ times the ${q^{th}}$ term of an $A.P.$, then ${(p + q)^{th}}$ term is