Careers360 Logo
JEE Main Registration 2025 Session 1, 2 - IIT Application Form Date, Fees, Documents Required

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.

Solve by difficulty

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:

 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:

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

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 :

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

"Stay in the loop. Receive exam news, study resources, and expert advice!"

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

E-books & Sample Papers

Get Answer to all your questions

Back to top