Harmonic Progression

The value of  $x_1{x}_2+x_2{x}_3+\dots +x_{n-1}{x}_n$ , when $x_1, x_2,\dots , x_n$ are in HP_____

A. $\left(n-1\right)\left(x_1-x_n\right)$

B. $\left(n-1\right)x_1{x}_n$

C. $n{x}_1{x}_n$

D. $n\left(x_1-x_n\right)$

If the roots of the equation $x\left(y-z\right)a^2+y\left(z-x\right)a+z\left(x-y\right)=0$ are equal, then $x, y$ and $z$ are in _____.

If $a,b$ and $c$ are in $\mathrm{HP}$, then correct statement is :

A. $a^2+c^2>b^2$

B. $a^2+c^2>2b^2$

C. $a^2+c^2<2b^2$

D. $a^2+c^2=2b^2$

We have two series, one is arithmetic progression, i.e., $9, 12, 15, \dots$ and another is Geometric Progression, i.e., $9, 27, 81, \dots .$. Now taking first three terms of each and adding its respective middle term to each term. Which series results into an Harmonic Progression?

$\mathrm{I}.$ Arithmetic progression

$\mathrm{II}.$ Geometric Progression

$\frac{1}{y-x}+\frac{1}{y-z}=\frac{1}{x}+\frac{1}{z}$, then $x, y, z$ are in ______.

How many Harmonic means can be inserted between $a$ and $b$ such that the product of $a$ and $b$ is $\frac{1}{20000}$ and the difference in the value of the variable $a$ and $b$ is $\frac{1}{200}$. The common difference for the corresponding Arithmetic Progression of the Harmonic Progression is $1$. (Take $a>b$)

The value of the common difference for the corresponding Arithmetic Progression of the Harmonic Progression was obtained for inserting $10$ harmonic mean between $\frac{1}{10}$ and $b$ is $\frac{1}{11}$, then find the value of $b$.

There are five harmonic mean is inserted between $a$ and $b$, then which of the following relation is used for finding the single harmonic mean between $a$ and $b$?

If the value of the single harmonic mean between $\frac{2}{22}$ and $\frac{2}{26}$ is $H_S$, and in the second case, if it was asked to insert the $3$ harmonic mean between $\frac{2}{22}$ and $\frac{2}{26}$, the value of the third harmonic mean between $\frac{2}{22}$ and $\frac{2}{26}$ is $\frac{2}{25}$. Find the value of the second harmonic mean.

If the value of the single harmonic mean between $\frac{1}{11}$ and $b$ is $H_S$, and in the second case, if it was asked to insert the $3$ harmonic mean between $\frac{1}{11}$ and $b$, the value of the second harmonic term between $a$ and $b$ is $\frac{1}{12}$, then find the value of $b$.

If the value of the single harmonic mean between $\frac{1}{11}$ and $\frac{1}{13}$ is $\frac{1}{12}$, and in the second case, if it was asked to insert the $3$ harmonic mean between $\frac{1}{11}$ and $\frac{1}{13}$ then find the value of common difference corresponding to arithmetic progression of harmonic progression.

How many Harmonic means can be inserted between $\frac{1}{100}$ and $\frac{1}{200}$ such that the common difference for the corresponding Arithmetic Progression of the Harmonic Progression is $1$?

Find the value of the common difference for the corresponding Arithmetic Progression of the Harmonic Progression obtained by inserting $10$ harmonic mean between $\frac{1}{10}$ and $\frac{1}{11}$.

Find $3$ harmonic mean between two $\frac{1}{11}$ and $\frac{1}{13}$.

Calculate the harmonic mean between $5 \mathrm{and} 9 ?$

If the $2^{nd} term \& 3^{rd} term$of a sequence are $1/14 \& 1/20$ respectively what would be sum of $7^{th} \& 9^{th} term$ of te HP.

If the sum of $2^{nd} \& 5^{th}$term of HP is $11/270 \& 6^{th}$term is $1/57.$ Find the $7^{th}$ term.

Which of the following sequence is a HP?

For a HP if the $3^{rd}$ term is $\frac{1}{23} \& 2nd term is \frac{1}{21.5}$ what is ${11}^{th}$ term?

What is the sum of the ${18}^{th}\& {15}^{th}$ term of the following series.

$\frac{1}{3},\frac{1}{7},\frac{1}{9},\frac{1}{11}$