**Test Yourself**

Page 296

1. The notation $\sum_{k = m}^{n}{a_k}$ is read "_____."

The summation from $k$ equals $m$ to $n$ of $a$ sub $k$.

2. The expanded form of $\sum_{k = m}^{n}{a_k}$ is _____.

$$ a_m + a_{m + 1} + a_{m + 2} + \dots + a_n $$

3. The value of $a_1 + a_2 + a_3 + \dots + a_n$ when $n = 2$ is "_____."

$$ a_1 + a_2 $$

4. The notation $\prod_{k = m}^{n}{a_k}$ is read "_____."

The product from $k$ equals $m$ to $n$ of $a$ sub $k$.

5. If $n$ is a positive integer, then $n! =$ _____.

$$ n \cdot (n - 1) \dots \cdot 3 \cdot 2 \cdot 1 $$

6. $\sum_{k = m}^{n}{a_k} + c\sum_{k = m}^{n}{b_k} =$ _____.

$$ \sum_{k = m}^{n}{a_k + cb_k} $$

7. $\left(\prod_{k = m}^{n}{a_k}\right)\left(\prod_{k = m}^{n}{b_k}\right) =$
   _____.

$$ \prod_{k = m}^{n}{a_kb_k} $$

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**Test Yourself**

Page 309

1. Mathematical induction is a method for proving that a property defined for
   integers $n$ is true for all values of $n$ that are _____.

greater than or equal to some initial value.

2. Let $P(n)$ be a property defined for integers $n$ and consider constructing a
   proof by mathematical induction for the statement "P(n) is true for all
   $n \geq a$."

a. In the basis step one must show _____.

that $P(a)$ is true.

b. In the inductive step one supposes that _____ for a particular but
arbitrarily chosen value of an integer $k \geq a$. This supposition is called
the _____. One then has to show that _____.

$P(k)$ is true; inductive hypothesis; $P(k + 1)$ is true.

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**Test Yourself**

Page 320

1. Mathematical induction differs from the kind of induction used in the natural
   sciences because it is actually a form of _____ reasoning.

deductive

2. Mathematical induction can be used to _____ conjectures that have been made
   using inductive reasoning.

prove

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**Test Yourself**

Page 333

1. In a proof by strong mathematical induction the basis step may require
   checking a property $P(n)$ for more _____ value of $n$.

than one

2. Suppose that in the basis step for a proof by strong mathematical induction
   the property $P(n)$ was checked for every integer $n$ from $a$ through $b$.
   Then in the inductive step one assumes that for any integer $k \geq b$, the
   property $P(n)$ is true for all values of $i$ from _____ through _____ and
   one shows that _____ is true.

$a$; $k$; $P(k + 1)$

3. According to the well-ordering principle for the integers, if a set $S$ of
   integers contains at least _____ and if there is some integer that is less
   than or equal to every _____, then _____.

one integer; integer in $S$; $S$ contains a least element.