How to explain this basic summation rule: $\sum_{i = 1}^n c = n\cdot c$, where $c$ is a constant?

Question

The parallel rule for definite integrals , namely,

$$\int_a^b c = c(b - a)$$ where $c$ s a constant, is rather intuitive, due to the fact that this number is computed in a way similar to the area of a rectangle. But it is difficult to make sense of a sum involving a variable $i$ that is not a term of this sum, the only term being the constant $c$ (but I may miss something here).

Is the rule true by definition so to say, or can it be explained by another more basic rule?

Answer 1

It is a consequence of the distributive property. Note that

$$\begin{align*} \sum_{i=1}^n c &= \underbrace{c+c+\ldots+c}_{n\text{ times}} \\ &= \underbrace{1\cdot c+1\cdot c+\ldots+1\cdot c}_{n\text{ times}} \\ &= \left(\underbrace{1+1+\ldots+1}_{n\text{ times}}\right)\cdot c \\ &= n\cdot c \end{align*}$$

Answer 2

It's the same. $c=c\cdot 1$, the area of a rectangle of height $c$ and width one. The sum says add up $n$ of 'em.