Chapter 13

Use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers \(n\). $$ 2+4+6+\cdots+2 n=n(n+1) $$

For the function \(f(x)=\frac{x-1}{x},\) find \(f(2)\) and \(f(3)\)

Use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers \(n\). $$ 1+5+9+\cdots+(4 n-3)=n(2 n-1) $$

How much do you need to invest now at \(5 \%\) per annum compounded monthly so that in 1 year you will have $$\$ 10,000 ?

True or False A function is a relation between two sets \(D\) and \(R\) so that each element \(x\) in the first set \(D\) is related to exactly one element \(y\) in the second set \(R\)

Use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers \(n\). $$ 3+4+5+\cdots+(n+2)=\frac{1}{2} n(n+5) $$

Use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers \(n\). $$ 3+5+7+\cdots+(2 n+1)=n(n+2) $$

If \(|r|<1,\) the sum of the geometric series \(\sum_{k=1}^{\infty} a r^{k-1}\) is _____ .

True or False The sum \(S_{n}\) of the first \(n\) terms of an arithmetic sequence \(\left\\{a_{n}\right\\}\) whose first term is \(a_{1}\) is found using the formula \(S_{n}=\frac{n}{2}\left(a_{1}+a_{n}\right)\)

True or False The notation \(a_{5}\) represents the fifth term of a sequence.

Use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers \(n\). $$ 2+5+8+\cdots+(3 n-1)=\frac{1}{2} n(3 n+1) $$

If a series does not converge, it is called a(n) _____ series. (a) arithmetic (b) divergent (c) geometric (d) recursive

True or False If \(n \geq 2\) is an integer, then $$ n !=n(n-1) \cdots 3 \cdot 2 \cdot 1 $$

Use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers \(n\). $$ 1+4+7+\cdots+(3 n-2)=\frac{1}{2} n(3 n-1) $$

Multiple Choice If \(a_{n}=-2 n+7\) is the \(n\) th term of an arithmetic sequence, the first term is _____. (a) -2 (b) 0 (c) 5 (d) 7

Multiple Choice The sequence \(a_{1}=5, a_{n}=3 a_{n-1}\) is an example of a(n) _______ sequence. (a) alternating (b) recursive (c) Fibonacci (d) summation

Use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers \(n\). $$ 1+2+2^{2}+\cdots+2^{n-1}=2^{n}-1 $$

In Problems 7 -16, show that each sequence is arithmetic. Find the common difference, and list the first four terms. $$ \left\\{s_{n}\right\\}=\\{n+4\\} $$

The notation $$ a_{1}+a_{2}+a_{3}+\cdots+a_{n}=\sum_{k=1}^{n} a_{k} $$ is an example of _______ notation.

Use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers \(n\). $$ 1+3+3^{2}+\cdots+3^{n-1}=\frac{1}{2}\left(3^{n}-1\right) $$

For a geometric sequence with first term \(a_{1}\) and common ratio \(r,\) where \(r \neq 0, r \neq 1,\) the sum of the first \(n\) terms is \(S_{n}=a_{1} \cdot \frac{1-r^{n}}{1-r}\)

Show that each sequence is arithmetic. Find the common difference, and list the first four terms. $$ \left\\{s_{n}\right\\}=\\{n-5\\} $$

Multiple Choice \(\sum_{k=1}^{n} k=1+2+3+\cdots+n=\) ________. (a) \(n !\) (b) \(\frac{n(n+1)}{2}\) (c) \(n k\) (d) \(\frac{n(n+1)(2 n+1)}{6}\)

Use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers \(n\). $$ 1+4+4^{2}+\cdots+4^{n-1}=\frac{1}{3}\left(4^{n}-1\right) $$

Show that each sequence is geometric. Then find the common ratio and list the first four terms. $$ \left\\{s_{n}\right\\}=\left\\{4^{n}\right\\} $$

Evaluate each factorial expression. \(10 !\)

Show that each sequence is arithmetic. Find the common difference, and list the first four terms. $$ \left\\{a_{n}\right\\}=\\{2 n-5\\} $$

Use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers \(n\). $$ 1+5+5^{2}+\cdots+5^{n-1}=\frac{1}{4}\left(5^{n}-1\right) $$

Show that each sequence is geometric. Then find the common ratio and list the first four terms. $$ \left.s_{n}\right\\}=\left\\{(-5)^{n}\right\\} $$

Evaluate each factorial expression. \(9 !\)

Show that each sequence is arithmetic. Find the common difference, and list the first four terms. $$ \left\\{b_{n}\right\\}=\\{3 n+1\\} $$

Use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers \(n\). $$ \frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+\cdots+\frac{1}{n(n+1)} \equiv \frac{n}{n+1} $$

Show that each sequence is geometric. Then find the common ratio and list the first four terms. $$ \left\\{a_{n}\right\\}=\left\\{-3\left(\frac{1}{2}\right)^{n}\right\\} $$

Evaluate each factorial expression. \(\frac{9 !}{6 !}\)

Show that each sequence is arithmetic. Find the common difference, and list the first four terms. $$ \left\\{c_{n}\right\\}=\\{6-2 n\\} $$

Use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers \(n\). $$ \frac{1}{1 \cdot 3}+\frac{1}{3 \cdot 5}+\frac{1}{5 \cdot 7}+\cdots+\frac{1}{(2 n-1)(2 n+1)}=\frac{n}{2 n+1} $$

Evaluate each factorial expression. \(\frac{12 !}{10 !}\)

Show that each sequence is arithmetic. Find the common difference, and list the first four terms. $$ \left\\{a_{n}\right\\}=\\{4-2 n\\} $$

Use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers \(n\). $$ 1^{2}+2^{2}+3^{2}+\cdots+n^{2}=\frac{1}{6} n(n+1)(2 n+1) $$

Show that each sequence is geometric. Then find the common ratio and list the first four terms. $$ \left\\{c_{n}\right\\}=\left\\{\frac{2^{n-1}}{4}\right\\} $$

Evaluate each factorial expression. \(\frac{4 ! 11 !}{7 !}\)

Show that each sequence is arithmetic. Find the common difference, and list the first four terms. $$ \left\\{t_{n}\right\\}=\left\\{\frac{1}{2}-\frac{1}{3} n\right\\} $$

Use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers \(n\). $$ 1^{3}+2^{3}+3^{3}+\cdots+n^{3}=\frac{1}{4} n^{2}(n+1)^{2} $$

Show that each sequence is geometric. Then find the common ratio and list the first four terms. $$ \left\\{d_{n}\right\\}=\left\\{\frac{3^{n}}{9}\right\\} $$

Evaluate each factorial expression. \(\frac{5 ! 8 !}{3 !}\)

Use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers \(n\). $$ 4+3+2+\cdots+(5-n)=\frac{1}{2} n(9-n) $$

Show that each sequence is geometric. Then find the common ratio and list the first four terms. $$ \left\\{e_{n}\right\\}=\left\\{7^{n / 4}\right\\} $$

List the first five terms of each sequence. \(\left\\{s_{n}\right\\}=\\{n\\}\)

Show that each sequence is arithmetic. Find the common difference, and list the first four terms. $$ \left\\{s_{n}\right\\}=\left\\{\ln 3^{n}\right\\} $$

Use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers \(n\). $$ -2-3-4-\cdots-(n+1)=-\frac{1}{2} n(n+3) $$