The arithmetic mean is the average(mean) of a number. It is calculated by dividing the sum of all numbers by the number of terms. In real life, we use arithmetic means to get general ideas of the data and also to get the average of wages.
In this article, we will cover the concept of the Arithmetic mean. This category falls under the broader category of Sequence and series, which is a crucial Chapter in class 11 Mathematics. It is not only essential for board exams but also for competitive exams like the Joint Entrance Examination(JEE Main) and other entrance exams such as SRMJEE, BITSAT, WBJEE, BCECE, and more. Over the last ten years of the JEE Main exam (from 2013 to 2023), a total of ten questions have been asked on this concept, including two in 2014, three in 2017, one in 2019, one in 2020, one in 2021, and two in 2023.
JEE Main: Study Materials | High Scoring Topics | Preparation Guide
JEE Main: Syllabus | Sample Papers | Mock Tests | PYQs
An arithmetic progression is a sequence in which each term increases or decreases by a constant term or fixed number. This fixed number is called the common difference of an AP and is generally denoted by '
If
Then
In AP, the first term is generally denoted by ‘
Also,
In AP, the first term is generally denoted by 'a'
We found a formula for the general term of a sequence, we can also find a formula for the general term of an arithmetic sequence. First, write the first few terms of a sequence where the first term is 'a' and the common difference is '
i.e.
Then the nth term (general term) of the A.P. is
On simplification of the general term, we can see that the general term of an AP is always linear in
The sum of all numbers in a list divided by the number of terms in a list is called the Arithmetic Mean.
If three terms are in AP , then the middle term is called the Arithmetic Mean (A.M.) of the other two numbers. So if
If
Insertion of
If there are n arithmetic mean between two numbers then all the mean are in arithmetic progression (AP).
If
Let
Now,
The sum of the n arithmetic mean between two numbers is n times the single A.M. between them.
Let
Then,
Example 1: For three positive integers
Solution
Given,
Now
Now
Now
Hence the answer is 2.
Example 2 Let be in an arithmetic progression, with
Solution:
Given,
Mean
Lets find the common difference(d),
Hence the answer is 10049.50.
Example 3: If arithmetic means are inserted between a and 100 such that the ratio of the first mean to the last mean is
[JEE MAINS 2022]
Solution
Let the common difference of the AP obtained be
Also
Using (i)
Hence, the value of
Example 4 :The common difference of the
[JEE MAINS 2022]
Solution
Given,
Subtract :
Using equation (1)
Hence, the answer is - 81
Example 5: If
Solution:
Arithmetic mean of two numbers (AM)-
So,
Hence the value of
The sum of all numbers in a list divided by the number of terms in a list is called the Arithmetic Mean.
If
If
Yes, the Arithmetic mean can be negative. If we found the arithmetic mean of negative numbers it would be negative.
02 Jul'25 08:06 PM
02 Jul'25 08:06 PM
02 Jul'25 08:06 PM
02 Jul'25 07:45 PM
02 Jul'25 07:43 PM
02 Jul'25 07:40 PM