If ${a^2},\,{b^2},\,{c^2}$ be in $A.P.$, then $\frac{a}{{b + c}},\,\frac{b}{{c + a}},\,\frac{c}{{a + b}}$ will be in

The $p^{\text {th }}, q^{\text {th }}$ and $r^{\text {th }}$ terms of an $A.P.$ are $a, b, c,$ respectively. Show that $(q-r) a+(r-p) b+(p-q) c=0$

If $1,\;{\log _y}x,\;{\log _z}y,\; - 15{\log _x}z$ are in $A.P.$, then

If the sum of three consecutive terms of an $A.P.$ is $51$ and the product of last and first term is $273$, then the numbers are

If $a_1, a_2, a_3, …….$ are in $A.P.$ such that $a_1 + a_7 + a_{16} = 40$, then the sum of the first $15$ terms of this $A.P.$ is

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...........