If $\frac{1}{a}, \frac{1}{b}, \frac{1}{c}$ are in AP, prove that
(i) $\frac{(b+c)}{a}, \frac{(c+a)}{b}, \frac{(a+b)}{c}$ are in AP.
(ii) $\frac{(b+c-a)}{a}, \frac{(c+a-b)}{b}, \frac{(a+b-c)}{c}$ are in AP.
(i) $\frac{(b+c)}{a}, \frac{(c+a)}{b}, \frac{(a+b)}{c}$ are in A.P.
To prove: $\frac{(b+c)}{a}, \frac{(c+a)}{b}, \frac{(a+b)}{c}$ are in A.P.
Given: $\frac{1}{a}, \frac{1}{b}, \frac{1}{c}$ are in A.P.
Proof: $\frac{1}{a}, \frac{1}{b}, \frac{1}{c}$ are in A.P.
If each term of an A.P. is multiplied by a constant, then the resulting sequence is also an A.P.
Multiplying the A.P. with (a + b + c )
$\Rightarrow \frac{(a+b+c)}{a}, \frac{(a+b+c)}{b}, \frac{(a+b+c)}{c}$ are also in A.P.
If a constant is subtracted from each term of an A.P., the resulting sequence is also an A.P.
Substracting the above A.P. with 1
$\Rightarrow \frac{(a+b+c)}{a}-1, \frac{(a+b+c)}{b}-1, \frac{(a+b+c)}{c}-1$, are also in A.P.
$\Rightarrow \frac{a+b+c-a}{a}, \frac{a+b+c-b}{b}, \frac{a+b+c-c}{c}$, are also in A.P.
$\Rightarrow \frac{b+c}{a}, \frac{a+c}{b}, \frac{a+b}{c}$, are also in A.P.
Hence Proved
(ii) $\frac{(b+c-a)}{a}, \frac{(c+a-b)}{b}, \frac{(a+b-c)}{c}$ are in A.P.
To prove: $\frac{(b+c-a)}{a}, \frac{(c+a-b)}{b}, \frac{(a+b-c)}{c}$ are in A.P.
Given: $\frac{1}{a}, \frac{1}{b}, \frac{1}{c}$ are in A.P.
Proof: $\frac{1}{a}, \frac{1}{b}, \frac{1}{c}$ are in A.P.
If each term of an A.P. is multiplied by a constant, then the resulting sequence is also an A.P.
Multiplying the A.P. with ( a + b + c )
$\Rightarrow \frac{(a+b+c)}{a}, \frac{(a+b+c)}{b}, \frac{(a+b+c)}{c}$ are also in A.P.
If a constant is subtracted from each term of an A.P., the resulting sequence is also an A.P.
Substracting the above A.P. with 2
$\Rightarrow \frac{(a+b+c)}{a}-2, \frac{(a+b+c)}{b}-2, \frac{(a+b+c)}{c}-2$, are also in A.P.
$\Rightarrow \frac{a+b+c-2 a}{a}, \frac{a+b+c-2 b}{b}, \frac{a+b+c-2 c}{c}$, are also in A.P.
$\Rightarrow \frac{b+c-a}{a}, \frac{a+c-b}{b}, \frac{a+b-c}{c}$, are also in A.P.
Hence Proved
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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,