Chapter 7.1, Problem 59E

Solution

To state: Whether the statement is true o false. If the statement is true, prove it. It the statement is false, give a counterexample.

  ${\displaystyle \sum _{i=1}^{n}k{a}_i}=k{\displaystyle \sum _{i=1}^n{a}_i}$ .

The statement is true.

Given information:

The given statement is ${\displaystyle \sum _{i=1}^{n}k{a}_i}=k{\displaystyle \sum _{i=1}^n{a}_i}$ .

Explanation:

Now consider the given statement,

  ${\displaystyle \sum _{i=1}^{n}k{a}_i}=k{\displaystyle \sum _{i=1}^n{a}_i}$

Let’s assume $k=3,a_i=2i,\text{and}n=5$ .

Substitute the values in the given summation notation,

  ${\displaystyle \sum _{i=1}^{5}3\left(2i\right)}=3{\displaystyle \sum _{i=1}^5\left(2i\right)}$

Evaluate ${\displaystyle \sum _{i=1}^{5}3\left(2i\right)}$ ,

  $\begin{array}{c}{\displaystyle \sum _{i=1}^{5}3\left(2i\right)}=3\left[2\left(1\right)\right]+3\left[2\left(2\right)\right]+3\left[2\left(3\right)\right]+3\left[2\left(4\right)\right]+3\left[2\left(5\right)\right]\\ =3\left(2\right)+3\left(4\right)+3\left(6\right)+3\left(8\right)+3\left(10\right)\\ =6+12+18+24+30\\ =90\end{array}$

Similarly, evaluate $3{\displaystyle \sum _{i=1}^5\left(2i\right)}$ .

  $\begin{array}{c}3{\displaystyle \sum _{i=1}^5\left(2i\right)}=3\left[2\left(1\right)+2\left(2\right)+2\left(3\right)+2\left(4\right)+2\left(5\right)\right]\\ =3\left[2+4+6+8+10\right]\\ =3\left(30\right)\\ =90\end{array}$

It can be seen that from both sides the resultant answer is 90.

Therefore, the given statement ${\displaystyle \sum _{i=1}^{n}k{a}_i}=k{\displaystyle \sum _{i=1}^n{a}_i}$ is true.