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Precalculus 7th

David Cohen, Theodore B. Lee, David Sklar

Chapter 14

Additional Topics in Algebra

Educators


Problem 1

Use the principle of mathematical induction to show that the statements are true for all natural numbers.
$$1+2+3+\cdots+n=n(n+1) / 2$$

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Problem 2

Use the principle of mathematical induction to show that the statements are true for all natural numbers.
$$2+4+6+\cdots+2 n=n(n+1)$$

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Problem 3

Use the principle of mathematical induction to show that the statements are true for all natural numbers.
$$1+4+7+\cdots+(3 n-2)=n(3 n-1) / 2$$

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Problem 4

Use the principle of mathematical induction to show that the statements are true for all natural numbers.
$$5+9+13+\dots+(4 n+1)=n(2 n+3)$$

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Problem 5

Use the principle of mathematical induction to show that the statements are true for all natural numbers.
$$1^{2}+2^{2}+3^{2}+\cdots+n^{2}=n(n+1)(2 n+1) / 6$$

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Problem 6

Use the principle of mathematical induction to show that the statements are true for all natural numbers.
$$2^{2}+4^{2}+6^{2}+\cdots+(2 n)^{2}=2 n(n+1)(2 n+1) / 3$$

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Problem 7

Use the principle of mathematical induction to show that the statements are true for all natural numbers.
$$1^{2}+3^{2}+5^{2}+\cdots+(2 n-1)^{2}=n(2 n-1)(2 n+1) / 3$$

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Problem 8

Use the principle of mathematical induction to show that the statements are true for all natural numbers.
$$2+2^{2}+2^{3}+\cdots+2^{n}=2^{n+1}-2$$

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Problem 9

Use the principle of mathematical induction to show that the statements are true for all natural numbers.
$$3+3^{2}+3^{3}+\cdots+3^{n}=\left(3^{n+1}-3\right) / 2$$

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Problem 10

Use the principle of mathematical induction to show that the statements are true for all natural numbers.
$$e^{x}+e^{2 x}+e^{3 x}+\cdots+e^{n x}=\frac{e^{(n+1) x}-e^{x}}{e^{x}-1} \quad(x \neq 0)$$

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Problem 11

Use the principle of mathematical induction to show that the statements are true for all natural numbers.
$$1^{3}+2^{3}+3^{3}+\cdots+n^{3}=[n(n+1) / 2]^{2}$$

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Problem 12

Use the principle of mathematical induction to show that the statements are true for all natural numbers.
$$2^{3}+4^{3}+6^{3}+\dots+(2 n)^{3}=2 n^{2}(n+1)^{2}$$

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Problem 13

Use the principle of mathematical induction to show that the statements are true for all natural numbers.
$$1^{3}+3^{3}+5^{3}+\cdots+(2 n-1)^{3}=n^{2}\left(2 n^{2}-1\right)$$

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Problem 14

Use the principle of mathematical induction to show that the statements are true for all natural numbers.
$$\begin{array}{r} 1 \cdot 2+3 \cdot 4+5 \cdot 6+\cdots+(2 n-1)(2 n) \\=n(n+1)(4 n-1) / 3\end{array}$$

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Problem 15

Use the principle of mathematical induction to show that the statements are true for all natural numbers.
$$\begin{array}{r}1 \cdot 3+3 \cdot 5+5 \cdot 7+\cdots+(2 n-1)(2 n+1) \\=n\left(4 n^{2}+6 n-1\right) / 3\end{array}$$

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Problem 16

Use the principle of mathematical induction to show that the statements are true for all natural numbers.
$$\begin{array}{r} \frac{1}{1 \times 3}+\frac{1}{2 \times 4}+\frac{1}{3 \times 5}+\dots+\frac{1}{n(n+2)} \\=\frac{n(3 n+5)}{4(n+1(n+2)}\end{array}$$

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Problem 17

Use the principle of mathematical induction to show that the statements are true for all natural numbers.
$$1+\frac{3}{2}+\frac{5}{2^{2}}+\frac{7}{2^{3}}+\dots+\frac{2 n-1}{2^{n-1}}=6-\frac{2 n+3}{2^{n-1}}$$

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Problem 18

Use the principle of mathematical induction to show that the statements are true for all natural numbers.
$$1+2 \cdot 2+3 \cdot 2^{2}+4 \cdot 2^{3}+\cdots+n \cdot 2^{n-1}=(n-1) 2^{n}+1$$

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Problem 19

Show that $n \leq 2^{n-1}$ for all natural numbers $n$

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Problem 20

Show that 3 is a factor of $n^{3}+2 n$ for all natural numbers $n$

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Problem 21

Show that $n^{2}+4<(n+1)^{2}$ for all natural numbers $n \geq 2$

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Problem 22

Show that $n^{3}>(n+1)^{2}$ for all natural numbers $n \geq 3$

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Problem 23

Prove that the statement is true for all natural numbers in the specified range. Use a calculator to carry out Step 1
$$(1.5)^{n}>2 \pi, n \geq 7$$

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Problem 24

Prove that the statement is true for all natural numbers in the specified range. Use a calculator to carry out Step 1
$$(1.25)^{n}>n, n \geq 11$$

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Problem 25

Prove that the statement is true for all natural numbers in the specified range. Use a calculator to carry out Step 1
$$(1.1)^{n}>n, n \geq 39$$

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Problem 26

Prove that the statement is true for all natural numbers in the specified range. Use a calculator to carry out Step 1
$$(1.1)^{n}>5 n, n \geq 60$$

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Problem 27

$$\text { Let } f(n)=\frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+\cdots+\frac{1}{n(n+1)}$$
(a) Complete the following table. \begin{tabular}{llllll} \hline$n$ & 1 & 2 & 3 & 4 & 5 \\$f(n)$ & & & & & \\\hline\end{tabular}
(b) On the basis of the results in the table, what would you guess to be the value of $f(6) ?$ Compute $f(6)$ to see whether this is correct.
(c) Make a conjecture about the value of $f(n),$ and prove it using mathematical induction.

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Problem 28

Let $f(n)=\frac{1}{1 \cdot 3}+\frac{1}{3 \cdot 5}+\frac{1}{5 \cdot 7}+\cdots+\frac{1}{(2 n-1)(2 n+1)}$
(a) Complete the following table.
$$\begin{array}{lcccc}\hline n & 1 & 2 & 3 & 4 \\f(n) & & & & \\\hline\end{array}$$
(b) On the basis of the results in the table, what would you guess to be the value of $f(5) ?$ Compute $f(5)$ to see whether your guess is correct.
(c) Make a conjecture about the value of $f(n),$ and prove it using mathematical induction.

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Problem 29

Suppose that a function $f$ satisfies the following conditions:
$$\begin{array}{l}
(1)=1 \\f(n)=f(n-1)+2 \sqrt{f(n-1)}+1 \quad(n \geq 2)\end{array}$$
(a) Complete the table.
$$\begin{array}{llllll}\hline n & 1 & 2 & 3 & 4 & 5 \\f(n) & & & & & \\\hline\end{array}$$
(b) On the basis of the results in the table, what would you guess to be the value of $f(6) ?$ Compute $f(6)$ to see whether your guess is correct.
(c) Make a conjecture about the value of $f(n)$ when $n$ is a natural number, and prove the conjecture using mathematical induction.

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Problem 30

This exercise demonstrates the necessity of carrying out both Step 1 and Step 2 before considering an induction proof valid.
(a) Let $P_{n}$ denote the statement that $n^{2}+1$ is even. Check that $P_{1}$ is true. Then give an example showing that $P_{n}$ is not true for all $n$
(b) Let $Q_{n}$ denote the statement that $n^{2}+n$ is odd. Show that Step 2 of an induction proof can be completed in this case, but not Step 1

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Problem 31

A prime number is a natural number that has no factors other than itself and 1. For technical reasons, 1 is not considered a prime. Thus, the list of the first seven primes looks like this: $2,3,5,7,11,13,17 .$ Let $P_{n}$ be the statement that $n^{2}+n+11$ is prime. Check that $P_{n}$ is true for all values of $n$ less than $10 .$ Check that $P_{10}$ is false.

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Problem 32

Prove that if $x \neq 1$
$$1+2 x+3 x^{2}+\cdots+n x^{n-1}=\frac{1-x^{n}}{(1-x)^{2}}-\frac{n x^{n}}{1-x}$$
for all natural numbers $n$

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Problem 33

If $r \neq 1,$ show that
$$ 1+r+r^{2}+\cdots+r^{n-1}=\frac{r^{n}-1}{r-1}$$ for all natural numbers $n$

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Problem 34

Use mathematical induction to show that
$$ x^{n}-1=(x-1)\left(1+x+x^{2}+\cdots+x^{n-1}\right)$$ for all natural numbers $n$

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Problem 35

Prove that 5 is a factor of $n^{5}-n$ for all natural numbers $n \geq 2$

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Problem 36

Prove that 4 is a factor of $5^{n}+3$ for all natural numbers $n .$

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Problem 37

Prove that 5 is a factor of $2^{2 n+1}+3^{2 n+1}$ for all nonnegative integers $n$

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Problem 38

Prove that 8 is a factor of $3^{2 n}-1$ for all natural numbers $n .$

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Problem 39

Prove that 3 is a factor of $2^{n+1}+(-1)^{n}$ for all nonnegative integers $n$

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Problem 40

Prove that 6 is a factor of $n^{3}+3 n^{2}+2 n$ for all natural numbers $n$

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Problem 41

Use mathematical induction to show that $x-y$ is a factor of $x^{n}-y^{n}$ for all natural numbers $n .$ Suggestion for Step 2: Verify and then use the fact that
$$x^{k+1}-y^{k+1}=x^{k}(x-y)+\left(x^{k}-y^{k}\right) y$$

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Problem 42

Use mathematical induction to prove that the formulas hold for all natural numbers $n$.
$$\log _{10}\left(a_{1} a_{2} \ldots a_{n}\right)=\log _{10} a_{1}+\log _{10} a_{2}+\dots+\log _{10} a_{n}$$

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Problem 43

Use mathematical induction to prove that the formulas hold for all natural numbers $n$.
$$(1+p)^{n} \geq 1+n p, \text { where } p>-1$$

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