Chapter F, Problem 37E

To determine

To prove: The formula ${\displaystyle \sum _{i=1}^{n}c}=nc$.

Explanation of Solution

The left hand side of the formula can be written as ${\displaystyle \sum _{i=1}^{n}c}=c{\displaystyle \sum _{i=1}^{n}1}$.

The sum ${\displaystyle \sum _{i=1}^{n}1}$ is defined as, ${\displaystyle \sum _{i=1}^{n}1}=\underset{\text{n}times}{\underbrace{1+1+\cdots +1}}=n$.

Thus, ${\displaystyle \sum _{i=1}^{n}c}=c{\displaystyle \sum _{i=1}^{n}1}=nc$.

Hence the formula is proved.