Chapter 4: Q. 33 (page 325)

Write out each sum in Exercises 33 in expanded form, and then calculate the value of the sum.
$\sum_{k = 0}^{9} \sqrt{3 + \frac{1}{10} k} \cdot (\frac{1}{10})$

Short Answer

Expert verified

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

Step by step solution

Given information

The given summation is $\sum_{k = 0}^{9} \sqrt{3 + \frac{1}{10} k} \cdot (\frac{1}{10})$ .

Write the summation in expanded form and evaluate the sum.

Rewrite the given summation in expanded form.

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

Write the conclusion

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

The sum is $1.8545$ .

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Write out each sum in Exercises 33 in expanded form, and then calculate the value of the sum.
$\sum_{k = 0}^{9} \sqrt{3 + \frac{1}{10} k} \cdot (\frac{1}{10})$

Short Answer

Step by step solution

Given information

Write the summation in expanded form and evaluate the sum.

Write the conclusion

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Write out each sum in Exercises 33 in expanded form, and then calculate the value of the sum.∑k=093+110k·110

Short Answer

Step by step solution

Given information

Write the summation in expanded form and evaluate the sum.

Write the conclusion

One App. One Place for Learning.

Most popular questions from this chapter

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Study anywhere. Anytime. Across all devices.

Write out each sum in Exercises 33 in expanded form, and then calculate the value of the sum.
$\sum_{k = 0}^{9} \sqrt{3 + \frac{1}{10} k} \cdot (\frac{1}{10})$