Prove the following by using the principle of mathematical induction for all n ∈ N: 102n – 1 + 1 is divisible by 11.
Prove the following by using the principle of mathematical induction for all $n \in N: 10^{2 n-1}+1$ is divisible by 11
Let the given statement be $\mathrm{P}(n)$, i.e.,
$P(n): 10^{2 n-1}+1$ is divisible by $11 .$
It can be observed that $\mathrm{P}(n)$ is true for $n=1$ since $\mathrm{P}(1)=10^{2.1-1}+1=11$, which is divisible by 11 .
Let $\mathrm{P}(k)$ be true for some positive integer $k$, i.e.,
$10^{2 k-1}+1$ is divisible by 11
$\therefore 10^{2 k-1}+1=11 m$, where $m \in \mathbf{N}$
We shall now prove that P(k + 1) is true whenever P(k) is true.
Consider
$10^{2(k+1)-1}+1$
$=10^{2 h+2-1}+1$
$=10^{2 k+1}+1$
$=10^{2}\left(10^{2 k-1}+1-1\right)+1$
$=10^{2}\left(10^{2 k-1}+1\right)-10^{2}+1$
$=10^{2} .11 \mathrm{~m}-100+1 \quad[$ Using $(1)]$
$=100 \times 11 \mathrm{~m}-99$
$=11(100 m-9)$
$=11 r$, where $r=(100 m-9)$ is some natural number
Therefore, $10^{2(k+1)-1}+1$ is divisible by 11 .
Thus, P(k + 1) is true whenever P(k) is true.
Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., 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,