TBIL-Cal Sequence Formulas (SQ1)

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Section 8.1 Sequence Formulas (SQ1)

Learning Outcomes

Define and use explicit and recursive formulas for sequences.

Subsection 8.1.1 Activities

Activity 8.1.1.

Which of the following are sequences?

monthly gas bill
days in the year
how long you wash dishes
$1, 1, 2, 3, 5, 8, \dots$
how much you spend on groceries

Activity 8.1.2.

Consider the sequence

1, 2, 4, \dots .

(a)

Which of the choices below reasonably continues this sequence of numbers?

$7, 12, 24, \dots$
$7, 11, 16, \dots$
$8, 16, 32, \dots$
$1, 2, 4, \dots$
$7, 12, 20, \dots$

(b)

Where possible, find a formula that allows us to move from one term to the next one.

Remark 8.1.3.

As seen in the previous activity, having too few terms may prevent us from finding a unique way to continue creating a sequence of numbers. In fact, we need sufficiently many terms to uniquely continue a sequence of numbers (and how many terms is sufficient depends on which sequence of numbers you are trying to generate). Sometimes, we do not want to write out all of the terms needed to allow for this. Therefore, we will want to find short-hand notation that allows us to do so.

Definition 8.1.4.

A sequence is a list of real numbers. Let

a_{n}

denote the

n

th term in a sequence. We will use the notation

{a_{n}}_{n = 1}^{\infty} = a_{1}, a_{2}, \dots, a_{n}, \dots .

A general formula that indicates how to explicitly find the

n

-th term of a sequence is the closed form of the sequence.

Activity 8.1.5.

Consider the sequence

1, \frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \dots .

Which of the following choices gives a closed formula for this sequence? Select all that apply.

${{(\frac{1}{3})}^{n - 1}}_{n = 1}^{\infty}$
${{(\frac{1}{3})}^{n}}_{n = 1}^{\infty}$
${{(\frac{1}{3})}^{n - 1}}_{n = 2}^{\infty}$
${{(\frac{1}{3})}^{n + 1}}_{n = 0}^{\infty}$
${{(\frac{1}{3})}^{n}}_{n = 0}^{\infty}$

Activity 8.1.6.

Let

a_{n}

be the

n

th term in the sequence

{\frac{n + 1}{n}}_{n = 1}^{\infty} .

Which of the following terms corresponds to the

27^{t h}

term of this sequence?

$\frac{27}{26}$
$\frac{26}{27}$
$\frac{27}{28}$
$\frac{28}{27}$
$\frac{29}{28}$

Activity 8.1.7.

Let

a_{n}

be the

n

th term in the sequence

{\frac{n + 1}{n}}_{n = 2}^{\infty} .

Which of the following terms corresponds to the

27^{t h}

term of this sequence?

$\frac{27}{26}$
$\frac{26}{27}$
$\frac{27}{28}$
$\frac{28}{27}$
$\frac{29}{28}$

Activity 8.1.8.

Let

a_{n}

be the

n

th term in the sequence

1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \dots .

Identify the

81

st term of this sequence.

$\frac{1}{79}$
$\frac{1}{80}$
$\frac{1}{81}$
$\frac{1}{82}$
$\frac{1}{83}$

Activity 8.1.9.

Find a closed form for the sequence

0, 3, 8, 15, 24, \dots .

Activity 8.1.10.

Find a closed form for the sequence

\frac{12}{1}, \frac{16}{2}, \frac{20}{3}, \frac{24}{4}, \frac{28}{5}, \dots .

Activity 8.1.11.

Let

a_{n}

be the

n

th term in the sequence

1, 1, 2, 3, 5, 8, \dots .

Find a formula for

a_{n} .

Definition 8.1.12.

A sequence is recursive if the terms are defined as a function of previous terms (with the necessary initial terms provided).

Activity 8.1.13.

Consider the sequence defined by

a_{1} = 6

and

a_{k + 1} = 4 a_{k} - 7

for

k \geq 1 .

What are the first four terms?

Activity 8.1.14.

Consider the sequence

2, 7, 22, 67, 202, \dots .

Which of the following offers the best recursive formula for this sequence?

$a_{n + 1} = 3 a_{n} + 1$
$a_{1} = 2, a_{k} = 3 a_{k - 1} + 1$ for $k > 1$
$a_{1} = 2, a_{2} = 7, a_{k} = 3 a_{k - 1} + 1$ for $k > 2$

Activity 8.1.15.

Once more, consider the sequence

1, 1, 2, 3, 5, 8, \dots

from Activity 8.1.11. Suppose

a_{1} = 1

and

a_{2} = 1 .

Give a recursive formula for

a_{n}

for all

n \geq 3 .

Activity 8.1.16.

Give a recursive formula that generates the sequence

1, 2, 4, 8, 16, 32, \dots .

Activity 8.1.17.

(a)

Find the first 5 terms of the following sequence:

$a_{n} = 3 \cdot 2^{n} .$

(b)

Find a closed form for the following sequence:

$4, 5, 8, 13, 20, \dots$

(c)

Find a recursive form for the following sequence:

$- 3, 2, 7, 12, 17, \dots$

Activity 8.1.18.

(a)

Find the first 5 terms of the following sequence:

$a_{n} = 5 n + 4.$

(b)

Find a closed form for the following sequence:

$0, 1, 4, 9, 16, \dots$

(c)

Find a recursive form for the following sequence:

$2, - 1, \frac{1}{2}, - \frac{1}{4}, \frac{1}{8}, \dots$

Subsection 8.1.2 Videos

Figure 181. Video: Define and use explicit and recursive formulas for sequences

Subsection 8.1.3 Exercises

Exercises available at https://tbil.org/calculus/preview/exercises/#/bank/SQ1/