Why average until $n-1 \times n \simeq \sum^{n-1} + avg_{n-1}$?

Question

I'm newbie in math and I want to understand it:

let $A = \{a_1,...,a_n\}$

why $$(\frac{1}{n-1}\sum_{i=1}^{n-1}a_i)n \simeq (\sum_{i=1}^{n-1}a_i)+ (\frac{1}{n-1}\sum_{i=1}^{n-1}a_i)$$

Exists any theorem to help me to understand the average field in a deep way ?

Thanks so much!

Answer 1

Hint $$\dfrac{n}{n-1}=\dfrac{n-1+1}{n-1}=1+\dfrac{1}{n-1}$$