For all $n \in N$,which of the following is true: $\frac{3^n-1}{2} \geq$ ?

Let $S_{n} = \frac{1}{2} + \frac{1}{6} + \frac{1}{12} + \frac{1}{20} + \ldots$ up to $n$ terms. If the sum of the first six terms of an $A.P.$ with first term $-p$ and common difference $p$ is $\sqrt{2026 S_{2025}}$,then the absolute difference between the $20^{\text{th}}$ and $15^{\text{th}}$ terms of the $A.P.$ is

If $a, b, c$ are in $G.P.$ and $x, y$ are the arithmetic means between $a, b$ and $b, c$ respectively,then $\frac{a}{x} + \frac{c}{y}$ is equal to

If the $(m + 1)^{th}$,$(n + 1)^{th}$,and $(r + 1)^{th}$ terms of an arithmetic progression are in geometric progression,and $m, n, r$ are in harmonic progression,then what is the ratio of the common difference to the first term of the arithmetic progression?

If $m$ is the $A.M.$ of two distinct real numbers $l$ and $n$ $(l, n > 1)$ and $G_1, G_2, G_3$ are three geometric means between $l$ and $n$,then $G_1^4 + 2G_2^4 + G_3^4$ equals:

If the sum of the first $40$ terms of the series $3+4+8+9+13+14+18+19+\ldots$ is $(102)m$,then $m$ is equal to: