If a, b, c are in a geometric progression and | vedclass

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

Question

(B) Given that $a, b, c$ are in a geometric progression,we have $b^2 = ac$ $(1)$.Let $a^x = b^y = c^z = k$.Then $a = k^{1/x}, b = k^{1/y}, c = k^{1/z}

Vedclass · Accepted Answer

Harmonic Progression

Vedclass · Answer

(B) Given that $a, b, c$ are in a geometric progression,we have $b^2 = ac$ $(1)$.Let $a^x = b^y = c^z = k$.Then $a = k^{1/x}, b = k^{1/y}, c = k^{1/z}$.Substituting these values into equation $(1)$:$(k^{1/y})^2 = k^{1/x} \cdot k^{1/z}$$k^{2/y} = k^{1/x + 1/z}$Equating the exponents: $\frac{2}{y} = \frac{1}{x} + \frac{1}{z}$.This implies that $\frac{1}{x}, \frac{1}{y}, \frac{1}{z}$ are in an arithmetic progression.Therefore,$x, y, z$ are in a harmonic progression.

If $a, b, c$ are in a geometric progression and $a^x = b^y = c^z$,then in which progression are $x, y, z$?

Similar Questions

If $a, b, c$ are in harmonic progression,then the straight line $\frac{x}{a} + \frac{y}{b} + \frac{1}{c} = 0$ always passes through a fixed point. That point is:

The sixth $H.M.$ between $3$ and $\frac{6}{13}$ is

Five numbers are in $HP$. The middle term is $1$ and the ratio of the second and the fourth terms is $2:1$. Then, the sum of the first three terms 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 ${a_1}, {a_2}, {a_3}, \dots, {a_n}$ are in $H.P.$,then the expression ${a_1}{a_2} + {a_2}{a_3} + \dots + {a_{n - 1}}{a_n}$ is equal to:

Vedclass Test Series

Exam Paper Generator

Online Exam Module