Q. 33 Write out each sum in Exercises ... [FREE SOLUTION]

Chapter 4: Q. 33 (page 325)

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

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 ${鈭�}_{k = 0}^{9} \sqrt{3 + \frac{1}{10} k} 路 (\frac{1}{10})$ .

Write the summation in expanded form and evaluate the sum.

Rewrite the given summation in expanded form.

$\begin{array}{rcl} {鈭�}_{k = 0}^{9} \sqrt{3 + \frac{1}{10} k} 路 (\frac{1}{10}) & = & (\sqrt{3 + \frac{1}{10} (0)} 路 (\frac{1}{10}) + \sqrt{3 + \frac{1}{10} (1)} 路 (\frac{1}{10}) + \sqrt{3 + \frac{1}{10} (2)} 路 (\frac{1}{10}) + \sqrt{3 + \frac{1}{10} (3)} 路 (\frac{1}{10}) + \sqrt{3 + \frac{1}{10} (4)} 路 (\frac{1}{10}) + \sqrt{3 + \frac{1}{10} (5)} 路 (\frac{1}{10}) + \sqrt{3 + \frac{1}{10} (6)} 路 (\frac{1}{10}) + \sqrt{3 + \frac{1}{10} (7)} 路 (\frac{1}{10}) + \sqrt{3 + \frac{1}{10} (8)} 路 (\frac{1}{10}) + \sqrt{3 + \frac{1}{10} (9)} 路 (\frac{1}{10})) \\ = & (\sqrt{3} 路 (\frac{1}{10}) + \sqrt{3 + \frac{1}{10}} 路 (\frac{1}{10}) + \sqrt{3 + \frac{1}{5}} 路 (\frac{1}{10}) + \sqrt{3 + \frac{3}{10}} 路 (\frac{1}{10}) + \sqrt{3 + \frac{2}{5}} 路 (\frac{1}{10}) + \sqrt{3 + \frac{1}{2}} 路 (\frac{1}{10}) + \sqrt{3 + \frac{3}{5}} 路 (\frac{1}{10}) + \sqrt{3 + \frac{7}{10}} 路 (\frac{1}{10}) + \sqrt{3 + \frac{4}{5}} 路 (\frac{1}{10}) + \sqrt{3 + \frac{9}{10}} 路 (\frac{1}{10})) \\ = & (\frac{\sqrt{3}}{10} + \sqrt{\frac{31}{10}} 路 (\frac{1}{10}) + \sqrt{\frac{16}{5}} 路 (\frac{1}{10}) + \sqrt{\frac{33}{10}} 路 (\frac{1}{10}) + \sqrt{\frac{17}{5}} 路 (\frac{1}{10}) + \sqrt{\frac{7}{2}} 路 (\frac{1}{10}) + \sqrt{\frac{18}{5}} 路 (\frac{1}{10}) + \sqrt{\frac{37}{10}} 路 (\frac{1}{10}) + \sqrt{\frac{19}{5}} 路 (\frac{1}{10}) + \sqrt{\frac{39}{10}} 路 (\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.
${鈭�}_{k = 0}^{9} \sqrt{3 + \frac{1}{10} k} 路 (\frac{1}{10})$

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.鈭�k=093+110k路110

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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Most popular questions from this chapter

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