Topic 2.1

Change in Arithmetic and Geometric Sequences

Two new streaming channels launch in the same month. Ridge starts with 1,200 subscribers and gains 300 more every month. Nova starts much smaller, with only 256 subscribers, but every month its audience is 1.5 times what it was the month before. Ridge begins more than four and a half times bigger. Which channel is bigger after a year?

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01
A sequence has inputs 0, 1, 2, 3, …, so its graph is separate dots.
02
Arithmetic: add $d$ every step. Geometric: multiply by $r$ every step.

Practice 33 problems Next: Chapter 2.2 Change in Linear and Exponential Functions

Most people bet on Ridge. By the end of this note you will see why that bet loses, and why. Unit 2 is about the difference between two kinds of change: adding the same amount again and again, and multiplying by the same factor again and again. Sequences are the simplest place to see that difference, so that is where we begin.

Remember from 1.3: a linear function has a constant rate of change. In this note you will meet its step-by-step cousin (the arithmetic sequence) and a new kind of pattern whose change is not constant but proportional (the geometric sequence).

What Is a Sequence?

CONCEPT

A Sequence Is a Function on Whole Numbers

A sequence is an ordered list of numbers. Each number is a term. We name the terms $a_{0}$ , $a_{1}$ , $a_{2}$ , … (read “a sub zero”, “a sub one”, …), and the small number $n$ in $a_{n}$ (the index) tells you the position. The index is a LABEL for the position, not a multiplication: $a_{3}$ is the term in position 3, not a times 3.

Think of a sequence as a function: the input is the index $n = 0$ , 1, 2, 3, …, and the output is the term $a_{n}$ .

Only whole-number inputs exist. There is a month 3 and a month 4, but no month 3.5. The outputs, though, can be any real number: negative, fractional, or decimal.

For example, the rule $t_{n} = n^{2} + 2$ gives $t_{0} = 2$ , $t_{1} = 3$ , and $t_{4} = 18$ . Asking for $t_{2.5}$ makes no sense, because 2.5 is not a position in the list. That is why the graph of a sequence is a set of separate dots. We never join them into a line or a curve.

KEY RULE

A sequence has inputs 0, 1, 2, 3, …, so its graph is separate dots.

This course focuses on two special kinds of sequences. Our two channels are one of each.

Arithmetic Sequences: Add the Same Amount

Here is Ridge's subscriber count at the end of each month ( $n = 0$ is launch month):

n (month)	0	1	2	3	4	5
$a_{n}$ (subscribers)	1200	1500	1800	2100	2400	2700
change	—	+300	+300	+300	+300	+300

CONCEPT

Common Difference $d$

A sequence is arithmetic when every step adds the same number $d$ . That number is the common difference.

Because the change over every step of the same length is constant, an arithmetic sequence has a constant rate of change, exactly like a linear function. For Ridge, $d = 300$ subscribers per month.

Why does the formula look the way it does? To get from $a_{0}$ to $a_{n}$ you take $n$ steps, and each step adds $d$ :

a_n = a_0 + (d + d + \cdots + d) = a_0 + d\,n \qquad (n\text{ copies of }d)

So the explicit rule, the rule that jumps straight to any term, is

a_n = a_0 + d\,n

For Ridge, $a_{0} = 1200$ and $d = 300$ , so $a_{n} = 1200 + 300n$ .

Now you can answer questions without listing every month. After one year, $a_{12} = 1200 + 300(12) = 4,800$ subscribers. After two years, $a_{24} = 8,400$ .

Geometric Sequences: Multiply by the Same Factor

Now look at Nova. This time the changes are not constant; they keep getting bigger. But divide each term by the one before it, and the answer is 1.5 every time.

n (month)	0	1	2	3	4	5
$g_{n}$ (subscribers)	256	384	576	864	1296	1944
change	—	+128	+192	+288	+432	+648
ratio	—	×1.5	×1.5	×1.5	×1.5	×1.5

CONCEPT

Common Ratio $r$

A sequence is geometric when every step multiplies by the same nonzero number $r$ . That number is the common ratio.

Put another way, over every step of the same length the output changes by the same proportion. For Nova, $r = 1.5$ means each month's audience is 150% of the previous month's, a 50% increase every month.

The formula comes from counting steps again. To get from $g_{0}$ to $g_{n}$ you multiply by $r$ a total of $n$ times:

g_n = g_0 \cdot (r \cdot r \cdots r) = g_0 \cdot r^{n} \qquad (n\text{ copies of }r)

So the explicit rule is

g_n = g_0 \cdot r^{n}

For Nova, $g_{0} = 256$ and $r = 1.5$ , so $g_{n} = 256(1.5)^{n}$ .

Left: every rise is +300. Right: every rise is 1.5 times the one before it, so the rises keep growing.

KEY RULE

Arithmetic: ADD $d$ every step. Geometric: MULTIPLY by $r$ every step.

Which Channel Wins?

Let's continue both tables past month 5:

n (month)	5	6	7	8
Ridge $a_{n}$	2700	3000	3300	3600
Nova $g_{n}$	1944	2916	4374	6561

Ridge is ahead through month 6. Nova passes it in month 7 and pulls away fast.

Ridge's gain is stuck at 300 every month. Nova's gain is 1.5 times its previous gain, so it grows every month: +128, +192, +288, and then +432 from month 3 to month 4. From that step on, Nova gains more than 300 each month and closes the gap, until it passes Ridge in month 7 and pulls away.

This is the big idea of Unit 2. Repeated multiplication by a factor greater than 1 eventually beats repeated addition, no matter how big a head start the adding side gets. In 2.2 you will see the same contest between linear and exponential functions.

Arithmetic, Geometric, or Neither?

CONCEPT

The Two Tests

Subtract consecutive terms. If every difference is the same, the sequence is arithmetic.

Divide consecutive terms. If every ratio is the same, the sequence is geometric.

If neither test gives a constant, the sequence is neither.

Worked example

Classify each sequence. (a) 7, 3, −1, −5, … (b) 48, −24, 12, −6, … (c) 2, 6, 12, 20, …

01
(a) Differences: $3 - 7 = -4$ , $-1 - 3 = -4$ , $-5 - (-1) = -4$ . The difference is always −4, so the sequence is arithmetic with $d = -4$ . A decreasing arithmetic sequence simply adds a negative number.
02
(b) Differences: −72, 36, −18. Not constant. Ratios: $-24 \div 48 = -1/2$ , $12 \div (-24) = -1/2$ , $-6 \div 12 = -1/2$ . Geometric with $r = -1/2$ . A negative ratio makes the terms switch sign every step.
03
(c) Differences: 4, 6, 8. Ratios: 3, 2, $5/3$ . Neither test gives a constant, so the sequence is neither arithmetic nor geometric.

COMMON MISTAKE

“It's going down, so it must be arithmetic.”

The direction of change tells you nothing about the type. In the example above, 7, 3, −1, −5 decreases and is arithmetic, while 48, −24, 12, −6 shrinks toward 0 and is geometric. Always run both tests.

“Nova's differences 128, 192, 288 keep changing, so there is no pattern.”

Growing differences are exactly what a geometric sequence with $r > 1$ produces. The pattern is in the ratios, not the differences.

Start From Any Term

The rules $a_{n} = a_{0} + dn$ and $g_{n} = g_{0} \cdot r^{n}$ start counting at term 0. But you can start from any term you know. If you know the term $a_{k}$ , then getting to term $n$ takes $(n - k)$ steps:

a_n = a_k + d\,(n - k)

g_n = g_k \cdot r^{\,n-k}

CONCEPT

Why $(n - k)$ ?

$(n - k)$ counts steps, not terms. From $a_{3}$ to $a_{8}$ there are $8 - 3 = 5$ steps, so $d$ is added 5 times.

Check with Ridge: $a_{3} = 2100$ , so $a_{n} = 2100 + 300(n - 3) = 2100 + 300n - 900 = 1200 + 300n$ . Same rule as before.

Check with Nova: $g_{2} = 576$ , so $g_{n} = 576(1.5)^{n-2}$ . Since $576 \div 1.5^{2} = 256$ , this is the same as $256(1.5)^{n}$ .

COMMON MISTAKE

“Month 1 had 900 subscribers and the channel gains 150 per month, so $a_{n} = 900 + 150n$ .”

Here 900 is $a_{1}$ , not $a_{0}$ . With the anchor at $k = 1$ , the rule is $a_{n} = 900 + 150(n - 1)$ . Check $n = 4$ : the correct rule gives $900 + 150(3) = 1,350$ (the list is 900, 1050, 1200, 1350), but the wrong rule gives 1,500, one step too many.

The familiar formula $a_{n} = a_{1} + d(n - 1)$ is just the $k = 1$ case. Before writing any rule, check which index the first given term has, and subtract that index.

Quick check

An arithmetic sequence has $a_3 = 20$ and $d = 6$ . Using $a_3$ as the anchor, what is $a_{10}$ ?

Given Two Terms, Find the Rule

Two terms are enough to rebuild the whole sequence, because the gap between their indices tells you how many steps separate them.

Worked example

Example 1 (arithmetic). A sequence has $a_{4} = 17$ and $a_{11} = 52$ . Find a rule for $a_{n}$ and find $a_{20}$ .

01
Steps between the terms: $11 - 4 = 7$ . Total change: $52 - 17 = 35$ . Each step adds $d = 35 \div 7 = 5$ .
02
Anchor at $a_{4}$ : $a_{n} = 17 + 5(n - 4)$ . (The same rule anchored at 0: $a_{0} = 17 - 5 \cdot 4 = -3$ , so $a_{n} = -3 + 5n$ .)
03
$a_{20} = 17 + 5(20 - 4) = 17 + 80 = 97$ .

Worked example

Example 2 (geometric, odd number of steps). A sequence has $g_{2} = 250$ and $g_{5} = 16$ . Find a rule for $g_{n}$ and find $g_{7}$ .

01
From $n = 2$ to $n = 5$ is 3 steps, so $r$ is multiplied in 3 times: $250 \cdot r^{3} = 16$ , which gives $r^{3} = 16/250 = 8/125$ .
02
Take the cube root: $r = 2/5 = 0.4$ . An odd root of a real number has exactly one real answer, so this is the only possible ratio.
03
$g_{n} = 250(0.4)^{n-2}$ , so $g_{7} = 250(0.4)^{5} = 2.56$ . (Starting from $g_{5}$ instead: $16(0.4)^{2} = 2.56$ . Same answer.)

Worked example

Example 3 (geometric, even number of steps). A sequence has $g_{1} = 5$ and $g_{5} = 405$ . Find every possible rule, and find $g_{4}$ .

01
From $n = 1$ to $n = 5$ is 4 steps: $5 \cdot r^{4} = 405$ , so $r^{4} = 81$ .
02
An even power hides the sign: $3^{4} = 81$ and $(-3)^{4} = 81$ . Both values of $r$ work, and nothing in the problem rules one out.
$r^{4} = 81 \quad\Rightarrow\quad r = 3 \quad\text{or}\quad r = -3$
03
So two sequences fit. $g_{n} = 5(3)^{n-1}$ gives 5, 15, 45, 135, 405, and $g_{n} = 5(-3)^{n-1}$ gives 5, −15, 45, −135, 405. Therefore $g_{4} = 135$ or $g_{4} = -135$ .

Both sequences pass through the two given terms. They agree at $n = 1$ , 3, 5 and have opposite signs at $n = 2$ and $n = 4$ . (The dashed lines only guide the eye; a sequence has no values between the dots.)

Some questions still have a single answer. For example, $g_{7} = 5( \pm 3)^{6} = 3,645$ either way, because 6 multiplications by −3 contain an even number of negative signs.

COMMON MISTAKE

“ $r^{4} = 81$ , so $r = 3$ .”

Keeping only the positive root throws away a whole valid sequence. An even root (square root, 4th root, …) of a positive number always has two real answers, + and −. Write both, then reject one only when the problem gives a reason (see Section 8).

“ $r^{4} = 81$ , so $r = 81 \div 4 = 20.25$ .”

The exponent 4 means multiplying $r$ by itself four times, not multiplying $r$ by 4. Check: $5 \cdot 20.25^{4} \approx 840,756$ , not 405.

Worked example

Try it yourself. (a) An arithmetic sequence has $a_{6} = -8$ and $a_{10} = 12$ . Find $a_{25}$ . (b) A geometric sequence has $g_{3} = 54$ and $g_{5} = 6$ . Find every possible value of $g_{6}$ .

Answers: (a) $d = 20 \div 4 = 5$ , so $a_{25} = -8 + 5(25 - 6) = 87$ . (b) $r^{2} = 6/54 = 1/9$ , so $r = 1/3$ or $r = -1/3$ . Then $g_{6} = 6 \cdot ( \pm 1/3) = 2$ or −2.

Quick check

A geometric sequence has $g_1 = 6$ and $g_4 = 162$ . What is the common ratio $r$ ?

Back to Nova: Context Decides

REAL-LIFE EXAMPLE

Nova's Missing Records

Suppose the only surviving records say Nova had 576 subscribers at the end of month 2 and 2,916 at the end of month 6, and that the growth is geometric. What is the monthly ratio, and in which month does Nova first pass 10,000 subscribers?

01
Month 2 to month 6 is 4 steps, an even number, so expect two candidates:

576\,r^{4} = 2916 \quad\Rightarrow\quad r^{4} = 5.0625 \quad\Rightarrow\quad r = \pm 1.5

02
Both values satisfy the equation, but the context rules one out. With $r = -1.5$ , month 3 would have $576(-1.5) = -864$ subscribers. A subscriber count can't be negative, so $r = 1.5$ .

03
Continue from $g_{2} = 576$ : month 9 gives $576(1.5)^{7} = 9,841.5$ and month 10 gives $576(1.5)^{8} = 14,762.25$ . Nova first passes 10,000 in month 10.

(A model can give decimals like 9,841.5; the real channel would have about 9,842.) In 2.13 you will solve $576(1.5)^{n-2} = 10,000$ directly with logarithms instead of building a table.

Practice

Worked example

P1. The terms 9, 14, 19, 24, … are labeled starting with $a_{1} = 9$ . Write an explicit rule and find $a_{40}$ .

Answer: $d = 5$ and the anchor is $a_{1}$ , so $a_{n} = 9 + 5(n - 1)$ . Then $a_{40} = 9 + 5(39) = 204$ .

Worked example

P2. The sequence 1000, 800, 640, … starts at $g_{0}$ . Is it arithmetic or geometric? Find $g_{6}$ .

Answer: The differences −200 and −160 are not equal, but the ratios are both 0.8. Geometric: $g_{n} = 1000(0.8)^{n}$ , so $g_{6} = 262.144$ .

Worked example

P3. A geometric sequence has $g_{2} = -12$ and $g_{5} = 96$ . Find $r$ and $g_{8}$ .

Answer: 3 steps, so $r^{3} = 96 \div (-12) = -8$ and $r = -2$ . A cube root has only one real answer, and it can be negative. Then $g_{8} = 96(-2)^{3} = -768$ .

Common slips

“It's going down, so it must be arithmetic.”

The direction of change tells you nothing about the type. In the example above, 7, 3, −1, −5 decreases and is arithmetic, while 48, −24, 12, −6 shrinks toward 0 and is geometric. Always run both tests.

“Nova's differences 128, 192, 288 keep changing, so there is no pattern.”

Growing differences are exactly what a geometric sequence with $r > 1$ produces. The pattern is in the ratios, not the differences.
“Month 1 had 900 subscribers and the channel gains 150 per month, so $a_{n} = 900 + 150n$ .”

Here 900 is $a_{1}$ , not $a_{0}$ . With the anchor at $k = 1$ , the rule is $a_{n} = 900 + 150(n - 1)$ . Check $n = 4$ : the correct rule gives $900 + 150(3) = 1,350$ (the list is 900, 1050, 1200, 1350), but the wrong rule gives 1,500, one step too many.

The familiar formula $a_{n} = a_{1} + d(n - 1)$ is just the $k = 1$ case. Before writing any rule, check which index the first given term has, and subtract that index.
“ $r^{4} = 81$ , so $r = 3$ .”

Keeping only the positive root throws away a whole valid sequence. An even root (square root, 4th root, …) of a positive number always has two real answers, + and −. Write both, then reject one only when the problem gives a reason (see Section 8).

“ $r^{4} = 81$ , so $r = 81 \div 4 = 20.25$ .”

The exponent 4 means multiplying $r$ by itself four times, not multiplying $r$ by 4. Check: $5 \cdot 20.25^{4} \approx 840,756$ , not 405.

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6 lines to re-read the night before.

01
A sequence is a function whose inputs are whole numbers, so its graph is separate dots.
02
Arithmetic: constant difference $d$ . $a_{n} = a_{0} + dn$ , or $a_{n} = a_{k} + d(n - k)$ . Constant rate of change, like a linear function.
03
Geometric: constant ratio $r$ . $g_{n} = g_{0} \cdot r^{n}$ , or $g_{n} = g_{k} \cdot r^{n-k}$ . Constant proportional change.
04
To classify, subtract consecutive terms (arithmetic test) and divide them (geometric test). Whether the terms go up or down does not decide the type.
05
$(n - k)$ counts steps. From two terms: change ÷ steps gives $d$ ; for $r$ , take the root that matches the number of steps. An even number of steps gives two candidates, $r$ and $-r$ : keep both unless the context rules one out.
06
Next, in 2.2: Ridge and Nova become functions of any real time $t$ , and we compare linear and exponential change directly.

Practice 33 problems Next: Chapter 2.2 Change in Linear and Exponential Functions

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Change in Arithmetic and Geometric Sequences

What Is a Sequence?

Arithmetic Sequences: Add the Same Amount

Geometric Sequences: Multiply by the Same Factor

Which Channel Wins?

Arithmetic, Geometric, or Neither?

Start From Any Term

Given Two Terms, Find the Rule

Back to Nova: Context Decides

Practice

Recap card

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