Question:
Find the sum to $n$ terms of the series $1 \times 2+2 \times 3+3 \times 4+4 \times 5+\ldots$
Solution:
The given series is $1 \times 2+2 \times 3+3 \times 4+4 \times 5+\ldots$
$n^{\text {th }}$ term, $a_{n}=n(n+1)$
$\therefore S_{n}=\sum_{k=1}^{n} a_{k}=\sum_{k=1}^{n} k(k+1)$
$=\sum_{k=1}^{n} k^{2}+\sum_{k=1}^{n} k$
$=\frac{n(n+1)(2 n+1)}{6}+\frac{n(n+1)}{2}$
$=\frac{n(n+1)}{2}\left(\frac{2 n+1}{3}+1\right)$
$=\frac{n(n+1)}{2}\left(\frac{2 n+4}{3}\right)$
$=\frac{n(n+1)(n+2)}{3}$
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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,
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