If the average of the first $n$ numbers in the sequence $148, 146, 144, \ldots$ is $125$,then $n$ is equal to

If the sum of the first $2n$ terms of the arithmetic progression $2, 5, 8, \dots$ is equal to the sum of the first $n$ terms of the arithmetic progression $57, 59, 61, \dots$,then $n = \dots$

Given that $n$ arithmetic means are inserted between two sets of numbers $(a, 2b)$ and $(2a, b)$, where $a, b \in \mathbb{R}$. Suppose the $m^{th}$ means between these sets are equal, then the ratio $a : b$ is equal to:

The sum of the first $p, q,$ and $r$ terms of an $A.P.$ are $a, b,$ and $c,$ respectively. Prove that $\frac{a}{p}(q-r)+\frac{b}{q}(r-p)+\frac{c}{r}(p-q)=0$.

If the sum of the first $21$ terms of the series $\log _{9^{1 / 2}} x + \log _{9^{1 / 3}} x + \log _{9^{1 / 4}} x + \dots$ where $x > 0$ is $504$,then $x$ is equal to:

Let $a_1=8, a_2, a_3, \ldots, a_n$ be an $A.P.$ If the sum of its first four terms is $50$ and the sum of its last four terms is $170$,then the product of its middle two terms is