57 lines
1.5 KiB
Markdown
57 lines
1.5 KiB
Markdown
**Test Yourself**
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Page 296
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1. The notation $\sum_{k = m}^{n}{a_k}$ is read "_____."
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The summation from $k$ equals $m$ to $n$ of $a$ sub $k$.
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2. The expanded form of $\sum_{k = m}^{n}{a_k}$ is _____.
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$$ a_m + a_{m + 1} + a_{m + 2} + \dots + a_n $$
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3. The value of $a_1 + a_2 + a_3 + \dots + a_n$ when $n = 2$ is "_____."
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$$ a_1 + a_2 $$
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4. The notation $\prod_{k = m}^{n}{a_k}$ is read "_____."
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The product from $k$ equals $m$ to $n$ of $a$ sub $k$.
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5. If $n$ is a positive integer, then $n! =$ _____.
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$$ n \cdot (n - 1) \dots \cdot 3 \cdot 2 \cdot 1 $$
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6. $\sum_{k = m}^{n}{a_k} + c\sum_{k = m}^{n}{b_k} =$ _____.
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$$ \sum_{k = m}^{n}{a_k + cb_k} $$
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7. $\left(\prod_{k = m}^{n}{a_k}\right)\left(\prod_{k = m}^{n}{b_k}\right) =$
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_____.
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$$ \prod_{k = m}^{n}{a_kb_k} $$
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---
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**Test Yourself**
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Page 309
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1. Mathematical induction is a method for proving that a property defined for
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integers $n$ is true for all values of $n$ that are _____.
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greater than or equal to some initial value.
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2. Let $P(n)$ be a property defined for integers $n$ and consider constructing a
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proof by mathematical induction for the statement "P(n) is true for all
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$n \geq a$."
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a. In the basis step one must show _____.
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that $P(a)$ is true.
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b. In the inductive step one supposes that _____ for a particular but
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arbitrarily chosen value of an integer $k \geq a$. This supposition is called
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the _____. One then has to show that _____.
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$P(k)$ is true; inductive hypothesis; $P(k + 1)$ is true.
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