Question:
4n – 1 is divisible by 3, for each natural number n.
Solution:
According to the question,
P(n) = 4n – 1 is divisible by 3.
So, substituting different values for n, we get,
P(0) = 40 – 1 = 0 which is divisible by 3.
P(1) = 41 – 1 = 3 which is divisible by 3.
P(2) = 42 – 1 = 15 which is divisible by 3.
P(3) = 43 – 1 = 63 which is divisible by 3.
Let P(k) = 4k – 1 be divisible by 3,
So, we get,
⇒ 4k – 1 = 3x.
Now, we also get that,
⇒ P(k+1) = 4k+1 – 1
= 4(3x + 1) – 1
= 12x + 3 is divisible by 3.
⇒ P(k+1) is true when P(k) is true
Therefore, by Mathematical Induction,
P(n) = 4n – 1 is divisible by 3 is true for each natural number n.
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- 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
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- Dual Nature of Radiation and Matter
- Atoms
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- Chemical Effects of Electric Current,
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