Sequences - Introduction to Relations and Functions (Pre-Algebra)

	$a_{1} > 0$	$a_{1} < 0$
$r > 1$	Increasing $3 \to \times 2 6 \to \times 2 12 \to \times 2 24 \to \times 2 48 \dots$	Decreasing $- 3 \to \times 2 - 6 \to \times 2 - 12 \to \times 2 - 24 \to \times 2 - 48 \dots$
$r = 1$	Constant $3 \to \times 1 3 \to \times 1 3 \to \times 1 3 \to \times 1 3 \dots$	Constant $- 3 \to \times 1 - 3 \to \times 1 - 3 \to \times 1 - 3 \to \times 1 - 3 \dots$
$0 < r < 1$	Decreasing $48 \to \times \frac{1}{2} 24 \to \times \frac{1}{2} 12 \to \times \frac{1}{2} 6 \to \times \frac{1}{2} 3 \dots$	Increasing $- 48 \to \times \frac{1}{2} - 24 \to \times \frac{1}{2} - 12 \to \times \frac{1}{2} - 6 \to \times \frac{1}{2} - 3 \dots$
$r < 0$	Alternating $3 \to \times (- 2) - 6 \to \times (- 2) 12 \to \times (- 2) - 24 \to \times (- 2) 48 \dots$	Alternating $- 3 \to \times (- 2) 6 \to \times (- 2) - 12 \to \times (- 2) 24 \to \times (- 2) - 48 \dots$

a_{1} > 0

a_{1} < 0

r > 1

Increasing

3 \to \times 2 6 \to \times 2 12 \to \times 2 24 \to \times 2 48 \dots

Decreasing

- 3 \to \times 2 - 6 \to \times 2 - 12 \to \times 2 - 24 \to \times 2 - 48 \dots

r = 1

Constant

$3 \to \times 1 3 \to \times 1 3 \to \times 1 3 \to \times 1 3 \dots$

Constant

$- 3 \to \times 1 - 3 \to \times 1 - 3 \to \times 1 - 3 \to \times 1 - 3 \dots$

0 < r < 1

Decreasing

48 \to \times \frac{1}{2} 24 \to \times \frac{1}{2} 12 \to \times \frac{1}{2} 6 \to \times \frac{1}{2} 3 \dots

Increasing

- 48 \to \times \frac{1}{2} - 24 \to \times \frac{1}{2} - 12 \to \times \frac{1}{2} - 6 \to \times \frac{1}{2} - 3 \dots

r < 0

Alternating

$3 \to \times (- 2) - 6 \to \times (- 2) 12 \to \times (- 2) - 24 \to \times (- 2) 48 \dots$

Alternating

$- 3 \to \times (- 2) 6 \to \times (- 2) - 12 \to \times (- 2) 24 \to \times (- 2) - 48 \dots$

Sequence	Classification
$15, 30, 45, 60, \dots$	Arithmetic Sequence

Sequence

Classification

15, 30, 45, 60, \dots

Arithmetic Sequence

Sequence	Classification
$15, 25, 15, 25, \dots$	Neither arithmetic nor geometric

Sequence

Classification

15, 25, 15, 25, \dots

Neither arithmetic nor geometric

Sequence	Classification
$5, 10, 20, 40, \dots$	Geometric Sequence

Sequence

Classification

5, 10, 20, 40, \dots

Geometric Sequence

a A sequence

a_{n}

can be considered a function of its position

n .

Recall that a term position takes whole number values. The terms of the given sequence can be organized in a table as follows.

In this sequence, the difference between consecutive terms is $7,$ so it is an arithmetic sequence with a common difference of $7 .$ This means that the fifth term of the sequence is calculated by adding $7$ to the fourth term.

To graph this sequence, start by drawing a coordinate plane where the horizontal axis represents the position $n$ and the vertical axis represents the term $a_{n} .$ Then, plot the ordered pairs $(n, a_{n})$ on the coordinate plane.

b Follow the same procedure as in Part A. First, make a table to show the terms of the given sequence.

In this sequence, the difference between the first and second terms is $6,$ but the difference between the second and third terms is $18 .$ There is no common difference between consecutive terms. However, there is a common ratio between consecutive terms — $3$ — so this is a geometric sequence. The fifth term of this sequence will be $3$ times the fourth term.

Looking over the completed graphs, Izabella said, The arithmetic sequence looks like a straight line, but the geometric sequence looks like a curve. I wonder if this will be important.

$n$	$a_{n}$	Rewrite
$1$	$a_{1}$	$a_{1} + 0 \cdot d$
$2$	$a_{2}$	$a_{1} + 1 \cdot d$
$3$	$a_{3}$	$a_{1} + 2 \cdot d$
$4$	$a_{4}$	$a_{1} + 3 \cdot d$
$5$	$a_{5}$	$a_{1} + 4 \cdot d$

n

a_{n}

Rewrite

1

a_{1}

a_{1} + 0 \cdot d

2

a_{2}

a_{1} + 1 \cdot d

3

a_{3}

a_{1} + 2 \cdot d

4

a_{4}

a_{1} + 3 \cdot d

5

a_{5}

a_{1} + 4 \cdot d

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Catch-Up and Review

Describing a Pattern

Sequence

Pattern

Numerical Patterns in Books

Answer

Hint

Solution

Finding the Terms of a Sequence

Special Sequences: Arithmetic Sequences

Arithmetic Sequence

Special Sequences: Geometric Sequences

Geometric Sequence

A Reading Challenge for the Mysterious Library

Hint

Solution

Identifying Sequences

Graphing Sequences

Answer

Hint

Solution

Explicit Rule of Arithmetic Sequences

Proof

Proof

Unlocking the Door

Hint

Solution

Identifying a Specific Term of an Arithmetic Sequence

The Fibonacci Sequence

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