How many terms of the AP 18 16, 14, 12, …. are needed to give the sum 78? Explain the double answer.
To Find: Number of terms required to make the sum 78.
Here $a=18, d=-2$
Let n be the number of terms required to make the sum 78.
$S_{n}=\frac{n}{2}[2 a+(n-1) d]$
$78=\frac{n}{2}[2 \times 18+(n-1)(-2)]$
$\Rightarrow 78 \times 2=36 n-2 n^{2}+2 n$
$\Rightarrow n^{2}-19 n+78=0$
$\Rightarrow n^{2}-6 n-13 n+78=0$
$\Rightarrow n(n-6)-13(n-6)=0$
$\Rightarrow(n-13)(n-6)=0$
Either n = 13 or n = 6
Explanation: Since the given $\mathrm{AP}$ is a decreasing progression where $\mathrm{a}_{\mathrm{n}-1}>\mathrm{a}_{\mathrm{n}}$, it is bound to have negative values in the series. $S_{n}$ is maximum for $n=9$ or $n=10$ since $T_{10}$ is $0\left(S_{10}=S_{9}=S_{\max }=90\right)$. The sum of 78 can be attained by either adding 6 terms or 13 terms so that negative terms from $T_{11}$ onward decrease the maximum sum to 78 .
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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,