Find the value of $n$ so that $\frac{a^{n+1}+b^{n+1}}{a^{n}+b^{n}}$ may be the geometric mean between $a$ and $b$.
G. M. of $a$ and $b$ is $\sqrt{a b}$.
By the given condition, $\frac{a^{n+1}+b^{n+1}}{a^{n}+b^{n}}=\sqrt{a b}$
Squaring both sides, we obtain
$\frac{\left(a^{n+1}+b^{n+1}\right)^{2}}{\left(a^{n}+b^{n}\right)^{2}}=a b$
$\Rightarrow a^{2 n+2}+2 a^{n+1} b^{n+1}+b^{2 n+2}=(a b)\left(a^{2 n}+2 a^{n} b^{n}+b^{2 n}\right)$
$\Rightarrow a^{2 n+2}+2 a^{n+1} b^{n+1}+b^{2 n+2}=a^{2 n+1} b+2 a^{n+1} b^{n+1}+a b^{2 n+1}$
$\Rightarrow a^{2 n+2}+b^{2 n+2}=a^{2 n+1} b+a b^{2 n+1}$
$\Rightarrow a^{2 n+2}-a^{2 n+1} b=a b^{2 n+1}-b^{2 n+2}$
$\Rightarrow a^{2 n+1}(a-b)=b^{2 n+1}(a-b)$
$\Rightarrow\left(\frac{a}{b}\right)^{2 n+1}=1=\left(\frac{a}{b}\right)^{0}$
$\Rightarrow 2 n+1=0$
$\Rightarrow n=\frac{-1}{2}$
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All Study Material
- JEE Main
- Exam Pattern
- Previous Year Papers
- PYQ Chapterwise
- Physics
- Kinematics 1D
- Kinemetics 2D
- Friction
- Work, Power, Energy
- Centre of Mass and Collision
- Rotational Dynamics
- Gravitation
- Calorimetry
- Elasticity
- Thermal Expansion
- Heat Transfer
- Kinetic Theory of Gases
- Thermodynamics
- Simple Harmonic Motion
- Wave on String
- Sound waves
- Fluid Mechanics
- Electrostatics
- Current Electricity
- Capacitor
- Magnetism and Matter
- Electromagnetic Induction
- Atomic Structure
- Dual Nature of Matter
- Nuclear Physics
- Radioactivity
- Semiconductors
- Communication System
- Error in Measurement & instruments
- Alternating Current
- Electromagnetic Waves
- Wave Optics
- X-Rays
- All Subjects
- Physics
- Motion in a Plane
- Law of Motion
- Work, Energy and Power
- Systems of Particles and Rotational Motion
- Gravitation
- Mechanical Properties of Solids
- Mechanical Properties of Fluids
- Thermal Properties of matter
- Thermodynamics
- Kinetic Theory
- Oscillations
- Waves
- Electric Charge and Fields
- Electrostatic Potential and Capacitance
- Current Electricity
- Thermoelectric Effects of Electric Current
- Heating Effects of Electric Current
- Moving Charges and Magnetism
- Magnetism and Matter
- Electromagnetic Induction
- Alternating Current
- Electromagnetic Wave
- Ray Optics and Optical Instruments
- Wave Optics
- Dual Nature of Radiation and Matter
- Atoms
- Nuclei
- Semiconductor Electronics: Materials, Devices and Simple Circuits.
- Chemical Effects of Electric Current,