Chapter 11.4, Problem 18WE

To determine

To calculate:

The following series with sigma notation:

  $\sqrt{7}+\sqrt{14}+\sqrt{21}+\mathrm{...}+\sqrt{77}$

Answer to Problem 18WE

The sigma notation for $\sqrt{7}+\sqrt{14}+\sqrt{21}+\mathrm{...}+\sqrt{77}$ is ${\displaystyle \sum _{n=1}^{11}\sqrt{7n}}$ .

Explanation of Solution

Given information:

  $\sqrt{7}+\sqrt{14}+\sqrt{21}+\mathrm{...}+\sqrt{77}$

Calculation:

The first term is $\sqrt{7}$ and the last term is $\sqrt{77}$ . Here, each term appears as the square root of the number $7$ as an index. Therefore, the summand is $\sqrt{7n}$ .

Now, the upper and lower limits are calculated as:

Use the first term to calculate the lower limit:

  $\begin{array}{l}\sqrt{7n}=\sqrt{7}\\ 7n=7\\ n=1\end{array}$

Use the last term to calculate the upper limit:

  $\begin{array}{l}\sqrt{7n}=\sqrt{77}\\ 7n=77\\ n=11\end{array}$

Now, the sigma notation for the given series is expressed as:

  $\sqrt{7}+\sqrt{14}+\sqrt{21}+\mathrm{...}+\sqrt{77}={\displaystyle \sum _{n=1}^{11}\sqrt{7n}}$