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Arithmetic Mean in AP - Practice Questions & MCQ

Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main

Quick Facts

  • Arithmetic Mean is considered one the most difficult concept.

  • 19 Questions around this concept.

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QuestionEASY

If 1, \log_{9}\left ( 3^{1-x}+2 \right ), \log_{3}\left ( 4.3^{x}-1 \right ) are in A.P. then x equals:

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 Let G be the geometric mean of two positive numbers a and b, and M be the arithmetic mean of  \frac{1}{a}  and \frac{1}{b}

if \frac{1}{M}:G  is 4:5 then a:b can be:

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For any three positive real numbers a, b and c, 9(25a2+b2)+25(c2−3ac)=15b(3a+c). Then:

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The number of ways in which 5 boys and 3 girls can be seated on a round table if a particular boy B1 and a particular girl G1 never sit adjacent to each other, is :

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Concepts Covered - 2

Arithmetic Mean

Arithmetic Mean

If three terms are in AP, then the middle term is called the Arithmetic Mean (A.M.) of other two numbers. So if, a, b and c are in A.P., then b is AM of a and c.

If a_1,a_2,a_3,.....,a_n are n positive numbers, then the Arithmetic Mean of these numbers is given by A=\frac{a_1+a_2+a_3+.....+a_n}{n} .

 

Insertion of n-Arithmetic Mean Between a and b

If \\\mathrm{\;A_1,A_2,A_3....,A_n} are n arithmetic mean between two numbers a and b, then, \mathrm{\mathit{a},A_1,A_2,A_3....,A_n,\mathit{b}} is an A.P.

Let d be the common difference of this A.P. Clearly, this A.P. contains n + 2 terms.

\\\mathrm{\therefore \mathit{b}=\mathit{(n+2)^{th}}\;term}\\\mathrm{\Rightarrow \mathit{b=a+((n+2)-1)d}=\mathit{a+(n+1)d}}\\\mathrm{\Rightarrow \mathit{d=\frac{b-a}{n+1}}}\\\\\mathrm{Now,}\\\mathrm{\mathit{A_1=a+d=\left ( a+\frac{b-a}{n+1} \right )}}\\\mathrm{\mathit{A_2=a+2d=\left ( a+\frac{2(b-a)}{n+1} \right )}}\\\mathrm{...}\\\mathrm{...}\\\mathrm{\mathit{A_n=a+nd=\left ( a+n\left (\frac{b-a}{n+1} \right ) \right )}}

Important Property of AM

The sum of n arithmetic mean between two numbers is n times the single A.M. between them.

Proof:

Let \\\mathrm{\;A_1,A_2,A_3....,A_n} be n arithmetic mean of two numbers a and b.

Then,\mathrm{\mathit{a},A_1,A_2,A_3....,A_n,\mathit{b}} is an A.P. with common difference \frac{b-a}{n+1} .

 

\\\mathrm{Now,}\\\mathrm{\;\;\;\;\;\;\;A_1+A_2+A_3+.......+A_n}\\\\\mathrm{\;\;\;\;\;\;\;=\frac{n}{2}\left [ A_1+A_n \right ]}\\\\\mathrm{\;\;\;\;\;\;\;=\frac{n}{2}\left [ \mathit{a+b} \right ]}\\\\\mathrm{\;\;\;\;\;\;\;[\because \mathit{a},A_1,A_2,....,A_n, b\;is\;an\;A.P.,\;\therefore \mathit{a+b=A_1+A_n}]}\\\\\mathrm{\;\;\;\;\;\;\;=n\left ( \mathit{\frac{a+b}{2}} \right )}\\\mathrm{\;\;\;\;\;\;\;=n\times(A.M.\;between\;\mathit{a}\;and\;\mathit{b})}

Study it with Videos

Arithmetic Mean
Important Property of AM

Study other Related Concepts

Arithmetic Mean in AP Current Topic

Sequence And Series
Arithmetic Progression
Sum of n Terms of an AP
Arithmetic Mean in AP
Geometric Progression
Geometric Mean In GP
Sum To n Terms Of a GP
Harmonic Progression
Harmonic Mean in HP
Relation between A.M., G.M. and H.M.

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Books

Reference Books

Arithmetic Mean

Mathematics for Joint Entrance Examination JEE (Advanced) : Algebra

Page No. : 5.8

Line : Last Line

Important Property of AM

Mathematics for Joint Entrance Examination JEE (Advanced) : Algebra

Page No. : 5.8

Line : Last Line

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