Q. 33 - Calculus

Question

Write out each sum in Exercises 33 in expanded form, and then calculate the value of the sum. 

$\sum _{k=0}^9\sqrt{3+\frac{1}{10}k}·\left(\frac{1}{10}\right)$

Step-by-Step Solution

Answer

The expanded notation is $\begin{array}{rcl}& & \frac{1}{10\sqrt{10}}\left(\sqrt{30}+\sqrt{31}+\sqrt{32}+\sqrt{33}+\sqrt{34}+\sqrt{35}+\sqrt{36}+\sqrt{37}+\sqrt{38}+\sqrt{39}\right)\end{array}$ and their sum is $1.8545$.

Step 1: Given information

The given summation is $\sum _{k=0}^9\sqrt{3+\frac{1}{10}k}·\left(\frac{1}{10}\right)$.

Step 2: Write the summation in expanded form and evaluate the sum.

Rewrite the given summation in expanded form. 

$$\begin{gathered} \begin{array}{rcl}\sum _{k=0}^9\sqrt{3+\frac{1}{10}k}·\left(\frac{1}{10}\right)& =& \left(\sqrt{3+\frac{1}{10}\left(0\right)}·\left(\frac{1}{10}\right)+\sqrt{3+\frac{1}{10}\left(1\right)}·\left(\frac{1}{10}\right)+\sqrt{3+\frac{1}{10}\left(2\right)}·\left(\frac{1}{10}\right)+\sqrt{3+\frac{1}{10}\left(3\right)}·\left(\frac{1}{10}\right)+\sqrt{3+\frac{1}{10}\left(4\right)}·\left(\frac{1}{10}\right) \\ +\sqrt{3+\frac{1}{10}\left(5\right)}·\left(\frac{1}{10}\right)+\sqrt{3+\frac{1}{10}\left(6\right)}·\left(\frac{1}{10}\right)+\sqrt{3+\frac{1}{10}\left(7\right)}·\left(\frac{1}{10}\right)+\sqrt{3+\frac{1}{10}\left(8\right)}·\left(\frac{1}{10}\right)+\sqrt{3+\frac{1}{10}\left(9\right)}·\left(\frac{1}{10}\right)\right)\\ & =& \left(\sqrt{3}·\left(\frac{1}{10}\right)+\sqrt{3+\frac{1}{10}}·\left(\frac{1}{10}\right)+\sqrt{3+\frac{1}{5}}·\left(\frac{1}{10}\right)+\sqrt{3+\frac{3}{10}}·\left(\frac{1}{10}\right)+\sqrt{3+\frac{2}{5}}·\left(\frac{1}{10}\right) \\ +\sqrt{3+\frac{1}{2}}·\left(\frac{1}{10}\right)+\sqrt{3+\frac{3}{5}}·\left(\frac{1}{10}\right)+\sqrt{3+\frac{7}{10}}·\left(\frac{1}{10}\right)+\sqrt{3+\frac{4}{5}}·\left(\frac{1}{10}\right)+\sqrt{3+\frac{9}{10}}·\left(\frac{1}{10}\right)\right)\\ & =& \left(\frac{\sqrt{3}}{10}+\sqrt{\frac{31}{10}}·\left(\frac{1}{10}\right)+\sqrt{\frac{16}{5}}·\left(\frac{1}{10}\right)+\sqrt{\frac{33}{10}}·\left(\frac{1}{10}\right)+\sqrt{\frac{17}{5}}·\left(\frac{1}{10}\right) \\ +\sqrt{\frac{7}{2}}·\left(\frac{1}{10}\right)+\sqrt{\frac{18}{5}}·\left(\frac{1}{10}\right)+\sqrt{\frac{37}{10}}·\left(\frac{1}{10}\right)+\sqrt{\frac{19}{5}}·\left(\frac{1}{10}\right)+\sqrt{\frac{39}{10}}·\left(\frac{1}{10}\right)\right)\\ & =& \left(\frac{\sqrt{3}}{10}+\frac{1}{10}\sqrt{\frac{31}{10}}+\frac{1}{10}\sqrt{\frac{16}{5}}+\frac{1}{10}\sqrt{\frac{33}{10}}+\frac{1}{10}\sqrt{\frac{17}{5}} \\ +\frac{1}{10}\sqrt{\frac{7}{2}}+\frac{1}{10}\sqrt{\frac{18}{5}}+\frac{1}{10}\sqrt{\frac{37}{10}}+\frac{1}{10}\sqrt{\frac{19}{5}}+\frac{1}{10}\sqrt{\frac{39}{10}}\right)\\ & =& \frac{1}{10}\left(\sqrt{3}+\sqrt{\frac{31}{10}}+\sqrt{\frac{16}{5}}+\sqrt{\frac{33}{10}}+\sqrt{\frac{17}{5}}+\sqrt{\frac{7}{2}}+\sqrt{\frac{18}{5}}+\sqrt{\frac{37}{10}}+\sqrt{\frac{19}{5}}+\sqrt{\frac{39}{10}}\right)\\ & =& \frac{1}{10}\left(\frac{\sqrt{30}+\sqrt{31}+\sqrt{32}+\sqrt{33}+\sqrt{34}+\sqrt{35}+\sqrt{36}+\sqrt{37}+\sqrt{38}+\sqrt{39}}{\sqrt{10}}\right)\\ & =& \frac{1}{10\sqrt{10}}\left(\sqrt{30}+\sqrt{31}+\sqrt{32}+\sqrt{33}+\sqrt{34}+\sqrt{35}+\sqrt{36}+\sqrt{37}+\sqrt{38}+\sqrt{39}\right)\\ & =& 0.0316\left(58.686\right)\\ & =& 1.8545\end{array} \end{gathered}$$

Step 3: Write the conclusion

The expanded form is $\begin{array}{rcl}& & \frac{1}{10\sqrt{10}}\left(\sqrt{30}+\sqrt{31}+\sqrt{32}+\sqrt{33}+\sqrt{34}+\sqrt{35}+\sqrt{36}+\sqrt{37}+\sqrt{38}+\sqrt{39}\right)\end{array}$.

The sum is $1.8545$.