How are this two summations are equal [Basic summations]

Can anyone explain to me how the following two summations are equal?

Answer

a rule that helps identify how to expand it from the first summation to the second summation

The basic rule that applies in these cases is that, if $a \le b \le c$ then:

$$
\sum_{j=a}^c f(j) = \sum_{j=a}^b f(j) + \sum_{j=b+1}^c f(j)

$$

The rule simply says that when you add the $c-a+1$ terms $f(a), f(a+1), \cdots,f(c)\,$, you get the same result if you split the sum in two, which follows directly from the associativity of addition:

$$f(a)+ f(a+1)+ \cdots+f(c) = \big(f(a) + f(a+1) + \cdots + f(b)\big) + \big(f(b+1) + \cdots + f(c)\big)$$

The rule can obviously be rewritten as:
$$
\sum_{j=b+1}^c f(j) = \sum_{j=a}^c f(j) - \sum_{j=a}^b f(j)
$$

The latter reduces to the posted question for $\,a=1, b=i, c=n\,$ and $\,f(j)=j\,$.