Harmonic Progression

Let $a_1, a_2, a_3,\dots .$ be in harmonic progression with $a_1=5$ and $a_{20}=25$. The least positive integer $n$ for which $a_n<0$ is

The corresponding first and the ${\left(2n-1\right)}^{\mathrm{th}}$ terms of an A.P., a G.P. and an H.P. are equal. If their $n^{\mathrm{th}}$ terms are $a,b$ and $c$ respectively, then

If $a,b,c$ are in A.P. and $a^2,b^2,c^2$ are in H.P. Then, which is of the following is/are possible?

If $a,b,c$ are three distinct numbers in G.P., $b,c,a$ are in A.P., and $a,bc,abc$ are in H.P., then the possible value of $b$ is

If $\frac{1}{b-a}+\frac{1}{b-c}=\frac{1}{a}+\frac{1}{c}$, then which of the following is/are possible?

If $A_1,A_2,G_1,G_2$ and $H_1 , H_2$ are two arithemetic, geometric and harmonic means, respectively, between two qualities $a$ and $b$, then $ab$ is equal to

If $x^2+9y^2+25z^2=xyz \left(\frac{15}{x}+\frac{5}{y}+\frac{3}{z}\right)$, then

If $p,q,$ and $r$ are in A.P. then which of the following is/are true?

If $a, b,$ and $c$ are in H.P.., then the value of $\frac{\left(ac+ab-bc\right) \left(ab+bc-ac\right)}{{\left(abc\right)}^2}$ is

Given that $x+y+z=15$ when $a, x, y,z ,b$ are in A.P. and $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{5}{3}$ when $a, x, y, z, b$ are in H.P. Then

If $a^2+2bc , b^2+2ca , c^2+2ab$ are in A.P. then

The number of positive integral ordered pairs of $\left(a,b\right)$ such that $6, a, b$ are in harmonic progression is

If $H_1,H_2,......,H_{20}$ are $20$ harmonic means between $2$ and $3$, then $\frac{H_1+2}{H_1-2}+\frac{H_{20}+3}{H_{20}-3}=$

If A.M., G.M., and H.M. of the first and last terms of the series $100, 101 , 102,....n-1, n$ are the terms of the series itself, then the value of $n$ is $\left(100 <n\le 500\right)$

Let $n\in N, n>25$. Let $A,G,H$ denote the arithmetic mean, geometric mean, and harmonic mean of $25$ and $n$. The least value of $n$ for which $A,G,H\in \left\{25, 26,....,n\right\}$ is

If $a,x$ and $b$ are in $\mathrm{AP}$, $a,y,$ and $b$ are in $\mathrm{GP}$, and $a, z, b$ are in $\mathrm{HP}$ such that $x=9z$ and $a>0, b>0$, then

If $a,b,$ and $c$ are in G.P. then $a+b, 2b,$ and $b+c$ are in

If in a progression $a_1,a_2,a_3,...,$ etc., $\left(a_r-a_{r+1}\right)$ bears a constant ratio with $a_r\times a_{r+1}$, then the terms of the progression are in

$a,b,c,d \in R^{+}$ such that $a,b,$ and $c$ are in A.P. and $b,c$ and $d$ are in H.P., then

If $a, b,$ and $c$ are in A.P., $p,q,$ and $r$ are in H.P., and $ap , bq,$ and $cr$ are in G.P. then $\frac{p}{r}+\frac{r}{p}$is equal to