
If \[\log (x + z) + \log (x + z - 2y) = 2\log (x - z)\],then \[x,y,z\] are in
A. H.P.
B. G.P.
C. A.P.
D. None of these
Answer
164.4k+ views
Hint
AP, GP, and horsepower symbolize the common or mean of the series. expected value, mean, and mean value, severally, square measure denoted by the letters AM, GM, and HM. The abbreviations AP, GP, and horsepower symbolize progression, progression, and patterned advance, severally. AN progression could be a set of integers with a set distinction between any 2 succeeding numbers.
By considering the reciprocals of the progression that doesn't contain zero, a patterned advance (HP) is outlined as a sequence of real numbers. The AP value per invoice is calculated by dividing the whole variety of invoices paid throughout a specified amount of your time by the whole prices incurred to pay those invoices.
Formula used:
If \[x,y,z\] are in AP
\[(y - x) = (z - y)\]
If \[x,y,z\] are in GP
\[{y^2} = xz\].
If \[x,y,z\] are in HP
\[\frac{1}{x} + \frac{1}{z} = \frac{2}{y}\]
\[\log (x z) = \log (x ) + \log (z)\]
Complete step-by-step solution
The given equation is
\[2\log (x - z) - \log (x - 2y + z) = \log (x + z)\]and
\[2\log (x - z) = \log (x - 2y + z) + \log (x + z)\]
So, \[{(x - z)^2} = (x + z)((x + z) - 2y)\]
This equation can also be written as
\[2y(x + z) = {(x + z)^2} - {(x - z)^2}\]
This equation is expanded as
\[2y(x + z) = (x + z + x - z)(x + z - x + z)\]
The equation is solved as
\[ = 2y(x + z) = 4xz\]
Hence, \[\frac{1}{x} + \frac{1}{z} = \frac{2}{y}\]
So, the H.P series has \[x,y,z\]
Therefore, the correct option is A.
Note
The first n terms of the arithmetic sequence are added together to form the sum of the first n terms of AP. AP, GP, and HP stand for the average or mean of the series. In mathematics, a set of numbers is referred to as an HP if the reciprocals of the terms are in AP. In a geometric progression (GP), the common ratio is multiplied by the previous term to produce each succeeding term.
AP, GP, and horsepower symbolize the common or mean of the series. expected value, mean, and mean value, severally, square measure denoted by the letters AM, GM, and HM. The abbreviations AP, GP, and horsepower symbolize progression, progression, and patterned advance, severally. AN progression could be a set of integers with a set distinction between any 2 succeeding numbers.
By considering the reciprocals of the progression that doesn't contain zero, a patterned advance (HP) is outlined as a sequence of real numbers. The AP value per invoice is calculated by dividing the whole variety of invoices paid throughout a specified amount of your time by the whole prices incurred to pay those invoices.
Formula used:
If \[x,y,z\] are in AP
\[(y - x) = (z - y)\]
If \[x,y,z\] are in GP
\[{y^2} = xz\].
If \[x,y,z\] are in HP
\[\frac{1}{x} + \frac{1}{z} = \frac{2}{y}\]
\[\log (x z) = \log (x ) + \log (z)\]
Complete step-by-step solution
The given equation is
\[2\log (x - z) - \log (x - 2y + z) = \log (x + z)\]and
\[2\log (x - z) = \log (x - 2y + z) + \log (x + z)\]
So, \[{(x - z)^2} = (x + z)((x + z) - 2y)\]
This equation can also be written as
\[2y(x + z) = {(x + z)^2} - {(x - z)^2}\]
This equation is expanded as
\[2y(x + z) = (x + z + x - z)(x + z - x + z)\]
The equation is solved as
\[ = 2y(x + z) = 4xz\]
Hence, \[\frac{1}{x} + \frac{1}{z} = \frac{2}{y}\]
So, the H.P series has \[x,y,z\]
Therefore, the correct option is A.
Note
The first n terms of the arithmetic sequence are added together to form the sum of the first n terms of AP. AP, GP, and HP stand for the average or mean of the series. In mathematics, a set of numbers is referred to as an HP if the reciprocals of the terms are in AP. In a geometric progression (GP), the common ratio is multiplied by the previous term to produce each succeeding term.
Recently Updated Pages
Environmental Chemistry Chapter for JEE Main Chemistry

Geometry of Complex Numbers – Topics, Reception, Audience and Related Readings

JEE Main 2021 July 25 Shift 1 Question Paper with Answer Key

JEE Main 2021 July 22 Shift 2 Question Paper with Answer Key

Difference Between Natural and Whole Numbers: JEE Main 2024

Hess Law of Constant Heat Summation: Definition, Formula & Applications

Trending doubts
JEE Main 2025 Session 2: Application Form (Out), Exam Dates (Released), Eligibility, & More

Atomic Structure - Electrons, Protons, Neutrons and Atomic Models

Displacement-Time Graph and Velocity-Time Graph for JEE

JEE Main 2025: Derivation of Equation of Trajectory in Physics

Learn About Angle Of Deviation In Prism: JEE Main Physics 2025

Electric Field Due to Uniformly Charged Ring for JEE Main 2025 - Formula and Derivation

Other Pages
JEE Advanced Marks vs Ranks 2025: Understanding Category-wise Qualifying Marks and Previous Year Cut-offs

JEE Advanced Weightage 2025 Chapter-Wise for Physics, Maths and Chemistry

NCERT Solutions for Class 11 Maths Chapter 4 Complex Numbers and Quadratic Equations

Degree of Dissociation and Its Formula With Solved Example for JEE

Instantaneous Velocity - Formula based Examples for JEE

JEE Main 2025: Conversion of Galvanometer Into Ammeter And Voltmeter in Physics
