If the p^{th},q^{th},and r^{th} terms of a ha | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) Let the harmonic progression be $\frac{1}{a}, \frac{1}{a+d}, \frac{1}{a+2d}, \dots$Then,$u = \frac{1}{a+(p-1)d}$,$v = \frac{1}{a+(q-1)d}$,and $w =

Vedclass · Accepted Answer

$0$

Vedclass · Answer

(B) Let the harmonic progression be $\frac{1}{a}, \frac{1}{a+d}, \frac{1}{a+2d}, \dots$Then,$u = \frac{1}{a+(p-1)d}$,$v = \frac{1}{a+(q-1)d}$,and $w = \frac{1}{a+(r-1)d}$.This implies $\frac{1}{u} = a+(p-1)d$,$\frac{1}{v} = a+(q-1)d$,and $\frac{1}{w} = a+(r-1)d$.Subtracting these equations,we get:$\frac{1}{u} - \frac{1}{v} = (p-q)d \implies \frac{v-u}{uv} = (p-q)d$$\frac{1}{v} - \frac{1}{w} = (q-r)d \implies \frac{w-v}{vw} = (q-r)d$$\frac{1}{w} - \frac{1}{u} = (r-p)d \implies \frac{u-w}{wu} = (r-p)d$Multiplying the terms,we observe that $(q-r)vw + (r-p)wu + (p-q)uv = 0$.

If the $p^{th}$,$q^{th}$,and $r^{th}$ terms of a harmonic progression are $u$,$v$,and $w$ respectively,then the value of $(q - r)vw + (r - p)wu + (p - q)uv$ is equal to:

Similar Questions

If $x, y, z$ are in $H.P.$,then the value of the expression $\log(x + z) + \log(x - 2y + z)$ is

If $a, b, c$ are in $H.P.$,then the value of $\left( \frac{1}{b} + \frac{1}{c} - \frac{1}{a} \right) \left( \frac{1}{c} + \frac{1}{a} - \frac{1}{b} \right)$ is

If the roots of the equation $x^3 - ax^2 + bx - c = 0$ are in harmonic progression $(HP)$,then the harmonic mean of the roots is

If $(y - x)$,$2(y - a)$,and $(y - z)$ are in $H.P.$,then $(x - a)$,$(y - a)$,and $(z - a)$ are in

If the roots of the equation $k x^3 - 18 x^2 - 36 x + 8 = 0$ are in harmonic progression,then $k =$

Vedclass Test Series

Exam Paper Generator

Online Exam Module