If the sum of the series $54 + 51 + 48 + .............$ is $513$, then the number of terms are

For $p, q \in R$, consider the real valued function $f ( x )=( x - p )^{2}- q , x \in R$ and $q >0$. Let $a _{1}, a _{2}, a _{3}$ and $a _{4}$ be in an arithmetic progression with mean $P$ and positive common difference. If $\left| f \left( a _{ i }\right)\right|=500$ for all $i=1,2,3,4$, then the absolute difference between the roots of $f ( x )=0$ is.

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

Insert $6$ numbers between $3$ and $24$ such that the resulting sequence is an $A.P.$

If $19^{th}$ terms of non -zero $A.P.$ is zero, then its ($49^{th}$ term) : ($29^{th}$ term) is

If three numbers be in $G.P.$, then their logarithms will be in