Arithmetic mean between 2 numbers are set of numbers such that all are in arithmetic progression.Therefore single arithmetic mean between two numbers a,b represents a number A such that a, A, b are in arithmetic progression and similarly n arithmetic mean between two numbers are set of n numbers such that they are in arithmetic progression
But this definition of arithmetic mean doesn’t apply to 3 or more than 3 numbers because if we apply the same definition it says arithmetic mean between 3 numbers(how is it possible one number between 3 numbers) represent a number such that all are in arithmetic progression!!!! How to apply this definition to 3 numbers
There is no such concept as "n arithmetic mean", you are misinformed.
One can indeed compute $m$ intermediate values forming an arithmetic progression between two given numbers by the interpolation formula $$\frac{ia+(m+1-i)b}{m+1}$$ where $i$ runs from $1$ to $m$. But there is no natural generalization to three numbers or more.
The arithmetic mean of $n$ numbers $a_k$ is the unique value$$\bar a=\frac1n\sum_{k=1}^n a_k.$$