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Chapter 10: Q. 8 (page 800)

In Exercises 3–8, let A, B, C, D, ... , Z be points in $R^{3}$ . Simplify the given quantity.
$\vec{A B} + \vec{B C} + . . . . . . + \vec{Y Z}$

Short Answer

Expert verified

$\vec{A B} + \vec{B C} + . . . . . . + \vec{Y Z} = \vec{A Z}$

Step by step solution

Given

$\vec{A B} + \vec{B C} + . . . . . . + \vec{Y Z}$

Simplify

From the law of vector addition:

\vec{A B} + \vec{B C} + . . . . . . + \vec{Y Z} = \vec{A Z}

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Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: The sum formulas in Theorem 4.4 can be applied only to sums whose starting index value is $k = 1$ .

(b) True or False: $\sum_{k = 0}^{n} \frac{1}{k + 1} + \sum_{k = 1}^{n} k^{2}$ is equal to $\sum_{k = 0}^{n} \frac{k^{3} + k^{2} + 1}{k + 1}$ .

(c) True or False: $\sum_{k = 1}^{n} \frac{1}{k + 1} + \sum_{k = 0}^{n} k^{2}$ is equal to $\sum_{k = 1}^{n} \frac{k^{3} + k^{2} + 1}{k + 1}$ .

(d) True or False: $(\sum_{k = 1}^{n} \frac{1}{k + 1}) (\sum_{k = 1}^{n} k^{2})$ is equal to $\sum_{k = 1}^{n} \frac{k^{2}}{k + 1}$ .

(e) True or False: $\sum_{k = 0}^{m} \sqrt{k} + \sum_{k = m}^{n} \sqrt{k}$ is equal to $\sum_{k = 0}^{n} \sqrt{k}$ .

(f) True or False: $\sum_{k = 0}^{n} a_{k} = - a_{0} - a_{n} + \sum_{k = 1}^{n - 1} a_{k}$ .

(g) True or False: ${(\sum_{k = 1}^{10} a_{k})}^{2} = \sum_{k = 1}^{10} a_{k}^{2}$ .

(h) True or False: $\sum_{k = 1}^{n} {(e^{x})}^{2} = \frac{e^{x} (e^{x} + 1) (2 e^{x} + 1)}{6}$ .

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In Exercises 3–8, let A, B, C, D, ... , Z be points in R3. Simplify the given quantity.AB→+BC→+......+YZ→

Short Answer

Step by step solution

Given

Simplify

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In Exercises 3–8, let A, B, C, D, ... , Z be points in $R^{3}$ . Simplify the given quantity.
$\vec{A B} + \vec{B C} + . . . . . . + \vec{Y Z}$