13.1 Sequences and Their Notations

Write the first five terms of the sequence whose general term is $a_{n} = 2 n - 7$ .

Write the first five terms of the sequence defined by the explicit formula $a_{n} = - 3 n + 8.$

Write the first five terms of the sequence defined by the explicit formula $t_{n} = 5 n - 4.$

Write the first five terms of the sequence.

Write the first five terms of the sequence.

Write the first six terms of the sequence.

Write the first six terms of the sequence.

Write an explicit formula for the $n \text{th}$ term of the sequence.

Write an explicit formula for the $n \text{th}$ term of the sequence.

Write an explicit formula for the $n \text{th}$ term of the sequence.

Write the first five terms of the sequence defined by the recursive formula.

Write the first 8 terms of the sequence defined by the recursive formula.

Write the first five terms of the sequence defined by the explicit formula $a_{n} = \frac{5 n}{\left(n + 2\right) !} .$

Write the first five terms of the sequence defined by the explicit formula $a_{n} = \frac{\left(n + 1\right) !}{2 n} .$

Discuss the meaning of a sequence. If a finite sequence is defined by a formula, what is its domain? What about an infinite sequence?

Is the ordered set of even numbers an infinite sequence? What about the ordered set of odd numbers? Explain why or why not.

What is a factorial, and how is it denoted? Use an example to illustrate how factorial notation can be beneficial.

Write the first four terms of the sequence.

$a_{n} = - \frac{16}{n + 1}$

Write the first four terms of the sequence.

$a_{n} = \frac{2^{n}}{n^{3}}$

Write the first four terms of the sequence.

$a_{n} = 1.25 \cdot \left(- 4\right)^{n - 1}$

Write the first four terms of the sequence.

$a_{n} = \frac{n^{2}}{2 n + 1}$

Write the first four terms of the sequence.

$a_{n} = - \left(\frac{4 \cdot \left(- 5\right)^{n - 1}}{5}\right)$

Write the first eight terms of the piecewise sequence.

$a_{n} = \left\{\right. \begin{matrix}\frac{n^{2}}{2 n + 1} & \text{if}\textrm{ } n \leq \text{5} \\ n^{2} - 5 & \text{if}\textrm{ } n >\text{5}\end{matrix}$

Write the first eight terms of the piecewise sequence.

$a_{n} = \left\{\right. \begin{matrix}- 0.6 \cdot 5^{n - 1} & \text{if}\textrm{ } n \text{is}\textrm{ }\text{prime}\textrm{ }\text{or}\textrm{ }\text{1} \\ 2.5 \cdot \left(- 2\right)^{n - 1} & \text{if}\textrm{ } n \text{is}\textrm{ }\text{composite}\end{matrix}$

Write an explicit formula for each sequence.

$4 , 7 , 12 , 19 , 28 , \ldots$

Write an explicit formula for each sequence.

$1 , 1 , \frac{4}{3} , 2 , \frac{16}{5} , \ldots$

Write an explicit formula for each sequence.

$1 , - \frac{1}{2} , \frac{1}{4} , - \frac{1}{8} , \frac{1}{16} , \ldots$

Write the first five terms of the sequence.

$a_{1} = 3 , a_{n} = \left(- 3\right) a_{n - 1}$

Write the first five terms of the sequence.

$a_{1} = - 1 , a_{n} = \frac{\left(- 3\right)^{n - 1}}{a_{n - 1} - 2}$

Write the first eight terms of the sequence.

$a_{1} = \frac{1}{24} , \left( \text{a}\right)_{2} = 1 , a_{n} = \left(2 a_{n - 2}\right) \left(3 a_{n - 1}\right)$

Write the first eight terms of the sequence.

$a_{1} = 2 , \left( \text{a}\right)_{2} = 10 , a_{n} = \frac{2 \left(a_{n - 1} + 2\right)}{a_{n - 2}}$

Write a recursive formula for each sequence.

$- 8 , - 6 , - 3 , 1 , 6 , \ldots$

Write a recursive formula for each sequence.

$35 , 38 , 41 , 44 , 47 , \ldots$

Evaluate the factorial.

$6 !$

Evaluate the factorial.

$\frac{12 !}{6 !}$

Write the first four terms of the sequence.

$a_{n} = \frac{n !}{n^{\text{2}}}$

Write the first four terms of the sequence.

$a_{n} = \frac{n !}{n^{2} - n - 1}$

Use the steps above to find the indicated term or terms for the sequence.

Find the first five terms of the sequence $a_{1} = \frac{87}{111}$, $a_{n} = \frac{4}{3} a_{n - 1} + \frac{12}{37} .$ Use the >Frac feature to give fractional results.

Use the steps above to find the indicated term or terms for the sequence.

Find the first five terms of the sequence $a_{1} = 2$, $a_{n} = 2^{\left[\left(a_{n} - 1\right) - 1\right]} + 1.$

Use the steps above to find the indicated term or terms for the sequence.

Find the tenth term of the sequence $a_{1} = 2$, $a_{n} = n a_{n - 1}$

Use the steps above to find the indicated terms for the sequence. Round to the nearest thousandth when necessary.

List the first six terms of the sequence $a_{n} = \frac{n^{3} - 3.5 n^{2} + 4.1 n - 1.5}{2.4 n} .$

Use the steps above to find the indicated terms for the sequence. Round to the nearest thousandth when necessary.

List the first four terms of the sequence $a_{n} = 5.7^{n} + 0.275 \left(n - 1\right) !$

Consider the sequence defined by $a_{n} = - 6 - 8 n .$ Is $a_{n} = - 421$ a term in the sequence? Verify the result.

Find a recursive formula for the sequence 1, 0, −1, −1, 0, 1, 1, 0, −1, −1, 0, 1, 1, .... (Hint: find a pattern for $a_{n}$ based on the first two terms.)

Prove the conjecture made in the preceding exercise.