1. Introduction
The Arithmetic Mean (AM) of two numbers is a value that comes exactly in between them when placed in an Arithmetic Progression (AP).
In simple words, the AM is the middle number between two values if they follow a constant difference.
Example: Between 6 and 14, the AM is 10 because 6, 10, 14 is an AP.
2. Meaning of Arithmetic Mean
If three numbers \(a, A, b\) form an AP, then:
\(A - a = b - A\)
Here, A is the Arithmetic Mean.
AM balances both sides equally, like the center point on a number line.
3. Formula for Arithmetic Mean Between Two Numbers
To find the AM between two numbers \(a\) and \(b\):
\(A = \dfrac{a + b}{2}\)
This gives the middle term that forms an AP with \(a\) and \(b\).
3.1. Example 1
Find the AM between 8 and 20.
\(A = \dfrac{8 + 20}{2} = 14\)
Sequence: 8, 14, 20
3.2. Example 2
Find the AM between -6 and 12.
\(A = \dfrac{-6 + 12}{2} = 3\)
Sequence: -6, 3, 12
4. Inserting Arithmetic Means Between Two Numbers
Sometimes we need to insert more than one AM between two numbers. When we insert n AMs between a and b, we create an AP with:
- First term \(= a\)
- Last term \(= b\)
- Total terms \(= n + 2\)
Then we use the nth term formula to find the common difference.
4.1. Example: Insert 3 AMs Between 2 and 14
Total terms = 3 AMs + 2 given numbers = 5
AP will be: 2, A1, A2, A3, 14
Use nth term formula:
\(a_5 = a + 4d\)
\(14 = 2 + 4d\)
\(12 = 4d\) → \(d = 3\)
So the AP is: 2, 5, 8, 11, 14
Inserted AMs: 5, 8, 11
4.2. Example: Insert 2 AMs Between 10 and 40
Total terms = 2 AMs + 2 = 4
AP: 10, A1, A2, 40
Use nth term:
\(40 = 10 + 3d\)
\(30 = 3d\) → \(d = 10\)
AP: 10, 20, 30, 40
Inserted AMs: 20, 30
5. Difference Between AM and Average
The AM between two numbers is the same as their simple average.
But AM between more than two numbers is different from the average concept studied in statistics.
6. Where the Arithmetic Mean Is Used
- To create smooth sequences
- To find missing middle values
- To divide a range evenly
- To check if three numbers form an AP
- To build AP-based problems in physics, economics, and finance
7. Common Mistakes
- Confusing AM with median (middle term in data).
- Incorrect formula use when inserting many AMs.
- Not counting total terms properly (n + 2 rule).
- Arithmetic errors when solving for d.
8. Quick Practice
Try these:
- Find the AM between 18 and 42.
- Insert 4 AMs between 5 and 30.
- Check if 9 is the AM of 6 and 12.
- The AM between two numbers is 25 and their difference is 20. Find the numbers.
9. Summary
- AM is the number that lies exactly between two numbers in an AP.
- Formula: \(A = \dfrac{a + b}{2}\)
- To insert n AMs: use AP logic with total terms = n + 2.
- AM helps divide numbers evenly and complete AP patterns.