Prove that the product of n geometric means between two quantities is equal to the nth power of a geometric mean of those two quantities.
Let $G_{1}, G_{2}, G_{3}, G_{4}, \ldots, G_{n}$ be $n$ G. M.s between $a$ and $b$.
Then, $a, G_{1}, G_{2}, G_{3}, G_{4}, \ldots, G_{n}, b$ is a G.P.
Let $r$ be the common ratio.
$\because b=a_{n+2}=a r^{(n+1)}$
$\Rightarrow r=\left(\frac{b}{a}\right)^{\frac{1}{(n+1)}}$
$\therefore G_{1}=a_{2}=a r$
$G_{2}=a_{3}=a r^{2}$
$G_{3}=a_{4}=a r^{3}$
$G_{n}=a_{(n+1)}=a r^{n}$
Also, let $G$ be the G.M. between $a$ and $b$.
$\therefore G^{2}=a b$
Now, $G_{1} \times G_{2} \times G_{3} \times G_{4} \times \ldots \times G_{n}=a r \times a r^{2} \times a r^{3} \times a r^{4} \times \ldots \times a r^{n}$
$=a^{n} \times r^{(1+2+3+4+\ldots \ldots+n)}$
$=a^{n} \times r^{\left(\frac{n(n+1)}{2}\right)}$
$=a^{n} \times\left[\left(\frac{b}{a}\right)^{\frac{1}{(n+1)}}\right]^{\left(\frac{n(n+1)}{2}\right)}$
$=a^{n} \times\left(\frac{b}{a}\right)^{\frac{n}{2}}$
$=a^{\frac{n}{2}} \times b^{\frac{n}{2}}$
$=(a b)^{\frac{n}{2}}$
$=(\sqrt{a b})^{n}$
$=G^{n}$
$\therefore G_{1} \times G_{2} \times G_{3} \times G_{4} \times \ldots \times G_{n}=G^{n}$
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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,