Arithmetic and Geometric Sequences

Part 1 · Hook

Why this matters

A coffee shop loyalty card adds 4 stamps per visit; a viral video's shares triple every hour. Both are lists of numbers in order, but one adds and the other multiplies, and after a few steps they look nothing alike.

Part 2 · Before you start

What this builds on

Part 3 · Prerequisite check

Quick check before you start

1. What is $2(3)^4$ ?

$162$
$1296$
$24$
$54$

Show the answer

Power first: $3^4 = 81$ , then $2 \cdot 81 = 162$ .

Correct: $162$ :
$1296$ :
$24$ :
$54$ :

2. What is the slope of the line through $(0, 5)$ and $(4, 17)$ ?

$3$
$4$
$12$
$\frac{1}{3}$

Show the answer

$\frac{17 - 5}{4 - 0} = 3$ .

Correct: $3$ :
$4$ :
$12$ :
$\frac{1}{3}$ :

Part 4 · See it

See it first

Dots at n = 0 to 5 for the arithmetic sequence a_n = 2 + 5n (2, 7, 12, 17, 22, 27) and the geometric sequence g_n = 2^n (1, 2, 4, 8, 16, 32); the geometric one passes the arithmetic one at n = 5. — An arithmetic sequence adds the same amount each step; a geometric sequence multiplies by the same amount. The geometric one starts lower and passes the arithmetic one at $n = 5$ . LevlPrep original diagram.

Part 5 · Step by step

How it works, step by step

Each term is the previous term plus $d$ an arithmetic sequence, $a_n = a_0 + dn$ , changing by equal amounts
Each term is the previous term times $r$ a geometric sequence, $g_n = g_0 r^n$ , changing by amounts that are multiplied by $r$
You know a term $a_k$ or $g_k$ that is not the firstuse $a_n = a_k + d(n - k)$ or $g_n = g_k r^{n - k}$

Part 6 · Key ideas

Key ideas

A sequence is a function whose inputs are whole numbers; its graph is a set of dots.
Arithmetic: common difference $d$ , so $a_n = a_0 + dn$ ; like a line sampled at whole numbers.
Geometric: common ratio $r$ , so $g_n = g_0 r^n$ . An increasing geometric sequence grows by increasing amounts.

Part 7 · Misconception

A common mistake

The wrong idea: If $g_1 = 48$ and $r = \frac{1}{2}$ , then $g_n = 48\left(\frac{1}{2}\right)^n$ .

What actually happens: The exponent counts steps from the known term: $g_n = 48\left(\frac{1}{2}\right)^{n - 1}$ , so that $g_1 = 48$ .

Part 8 · Check yourself

Check yourself

Exam-style questions. Anything you miss goes into your review queue.

Table

Two sequences

The first five terms of the sequences a and g, for n = 0 to 4
$n$	0	1	2	3	4
$a_n$	5	8	11	14	17
$g_n$	2	6	18	54	162

1. Which statement describes the two sequences in the table?

$a$ is geometric with common ratio $3$ ; $g$ is arithmetic with common difference $3$
$a$ is arithmetic with common difference $3$ ; $g$ is geometric with common ratio $3$
Both are arithmetic, with common differences $3$ and $4$
$a$ is arithmetic with common difference $5$ ; $g$ is geometric with common ratio $2$

Show the answer

Each term of $a$ is $3$ more than the one before ( $8 - 5 = 3$ , $11 - 8 = 3$ , ...). Each term of $g$ is $3$ times the one before ( $\frac{6}{2} = 3$ , $\frac{18}{6} = 3$ , ...).

$a$ is geometric with common ratio $3$ ; $g$ is arithmetic with common difference $3$ : The labels are swapped: $a$ adds $3$ , $g$ multiplies by $3$ .
Correct: $a$ is arithmetic with common difference $3$ ; $g$ is geometric with common ratio $3$ : Equal differences for $a$ ; equal ratios for $g$ .
Both are arithmetic, with common differences $3$ and $4$ : The differences of $g$ are $4, 12, 36, 108$ , which are not equal.
$a$ is arithmetic with common difference $5$ ; $g$ is geometric with common ratio $2$ : $5$ and $2$ are the first terms $a_0$ and $g_0$ , not the difference and ratio.

2. Using the table, find $a_{20}$ .

No calculator. Type the exact value (a fraction, π, e or a root is fine).

Show the answer

$a_n = a_0 + dn = 5 + 3n$ , so $a_{20} = 5 + 3(20) = 65$ .

Answer: 65

3. Using the table, find $g_6$ .

No calculator. Type the exact value (a fraction, π, e or a root is fine).

Show the answer

$g_n = g_0 r^n = 2(3)^n$ , so $g_6$ $= 2(3^6)$ $= 2(729)$ $= 1458$ .

Answer: 1458

Graph

A sequence shown as points

4. The graph shows the terms $h_1$ to $h_5$ of a geometric sequence. What is its common ratio?

No calculator. Type the exact value (a fraction, π, e or a root is fine).

Show the answer

$\frac{h_2}{h_1}$ $= \frac{24}{48}$ $= \frac{1}{2}$ , and every later ratio is also $\frac{1}{2}$ .

Answer: $\frac{1}{2}$ ≈ 0.500

5. Write $h_n$ in terms of $n$ , using the term $h_1 = 48$ .

Type an expression for $h_n =$ .

Show the answer

With a known term $h_k$ , $h_n = h_k r^{n - k}$ . Here $k = 1$ : $h_n = 48\left(\frac{1}{2}\right)^{n - 1}$ . Check: $h_3 = 48 \cdot \frac{1}{4} = 12$ .

Answer: $h_n =$ $48\,\left(\frac{1}{2}\right)^{n - 1}$

6. An arithmetic sequence has $a_0 = 7$ and common difference $-3$ . Find $a_{15}$ .

No calculator. Type the exact value (a fraction, π, e or a root is fine).

Show the answer

$a_{15}$ $= a_0$ $+ 15d$ $= 7$ $+ 15(-3)$ $= 7 - 45$ $= -38$ .

Answer: -38

Part 9 · Summary

Summary

Arithmetic sequences add a common difference:

a_n = a_k + d(n - k)

. Geometric sequences multiply by a common ratio:

g_n = g_k r^{n - k}

. Equal amounts versus multiplied amounts is the difference that grows into linear versus exponential change.

Part 10 · Up next

What comes next

Part 11 · Connections