If the $(m + 1)^{th}$,$(n + 1)^{th}$ and $(r + 1)^{th}$ terms of an $A.P.$ are in $G.P.$ and $m, n, r$ are in $H.P.$,then the value of the ratio of the common difference to the first term of the $A.P.$ is

Let $a_1, a_2, a_3, \dots$ be an $A$.$P$. and $g_1, g_2, g_3, \dots$ be an increasing $G$.$P$. If $a_1 = g_1$ and $a_2 + g_2 = 1$ and $a_3 + g_3 = 4$, then $a_{10} + g_5$ is equal to:

If three unequal numbers $p, q, r$ are in $H.P.$ and their squares are in $A.P.$,then the ratio $p:q:r$ 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

The sum of three consecutive terms in a geometric progression is $14$. If $1$ is added to the first and the second terms and $1$ is subtracted from the third,the resulting new terms are in arithmetic progression. Then the lowest of the original terms is

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