AP® Precalculus review sheet from Aim for Five (aimforfive.com/precalc/units/2/2-1)
Unit 2 · Topic 2.1
2.1 Change in Arithmetic and Geometric Sequences
A sequence is a list of numbers produced by a rule, one for each whole number. Arithmetic sequences add the same amount each step and geometric sequences multiply by the same amount. They are the stepping stones to linear and exponential functions.
Key terms
- sequence
- arithmetic sequence
- common difference
- geometric sequence
- common ratio
Sequences are functions
A sequence is a function whose inputs are whole numbers: 0, 1, 2, 3, and so on. You write the output for input n as aₙ, read “a sub n.” So a₀ is the first value, a₁ the next, and so on.
Because the inputs are separate whole numbers, the graph of a sequence is a set of separate dots, not a connected curve.
Some problems start counting at a₁ instead of a₀. Always check which index the first term has.
Arithmetic sequences
In an arithmetic sequence, you add the same number each step. That number is the common difference d. Example: 5, 8, 11, 14, … has d = 3.
Starting from the initial value a₀: aₙ = a₀ + dn.
Starting from any known term aₖ: aₙ = aₖ + d(n − k).
The common difference is a constant rate of change: each step of 1 in n changes the output by d.
Geometric sequences
In a geometric sequence, you multiply by the same number each step. That number is the common ratio r. Example: 3, 6, 12, 24, … has r = 2. Each term is a constant proportion of the one before, which is called constant proportional change.
Starting from the initial value g₀: gₙ = g₀ · rⁿ.
Starting from any known term gₖ: gₙ = gₖ · rⁿ⁻ᵏ.
If r > 1 (and the terms are positive), the sequence grows; if 0 < r < 1, it shrinks toward 0. A negative ratio makes the terms alternate in sign.
Reading sequences from graphs and tables
On a graph, the dots of an arithmetic sequence lie along a straight line, because each step of 1 to the right moves the dot up or down by the same amount d. The dots of a geometric sequence with positive terms and r > 1 curve upward more and more steeply. With 0 < r < 1, they drop toward the horizontal axis and flatten out.
In a table, check that n goes up by 1 before you compare neighboring terms. If n jumps by 2 and the terms show a constant difference D, then d = D/2. If they show a constant ratio R and the terms are positive, then r is the square root of R.
Telling them apart and comparing growth
To identify a sequence, check differences and ratios of consecutive terms. Constant differences mean arithmetic. Constant ratios mean geometric.
An increasing arithmetic sequence goes up by the same amount every step. An increasing geometric sequence with positive terms goes up by a larger amount every step, so it eventually passes any arithmetic sequence.
| Step n | Arithmetic: 10 + 20n | Geometric: 10 · 2ⁿ |
|---|---|---|
| 0 | 10 | 10 |
| 1 | 30 | 20 |
| 3 | 70 | 80 |
| 5 | 110 | 320 |
| 8 | 170 | 2,560 |
Worked examples
Try each one yourself first, then open the solution.
- Example 1
Arithmetic sequence from two terms
An arithmetic sequence has a₃ = 14 and a₈ = 39. Write a formula for aₙ and find a₂₀.
Show the solutionHide the solution
- Step 1: From term 3 to term 8 is 5 steps, and the value rises 39 − 14 = 25. So d = 25/5 = 5.
- Step 2: Using the known term a₃: aₙ = 14 + 5(n − 3).
- Step 3: Simplify if you like: aₙ = 14 + 5n − 15 = −1 + 5n, so a₀ = −1.
- Step 4: a₂₀ = −1 + 5(20) = 99.
Answer: aₙ = 14 + 5(n − 3) = 5n − 1; a₂₀ = 99.
- Example 2
Geometric sequence from two terms
A geometric sequence with positive terms has g₂ = 12 and g₅ = 96. Write a formula for gₙ and find g₈.
Show the solutionHide the solution
- Step 1: From term 2 to term 5 is 3 steps, so the value is multiplied by r three times: 12 · r³ = 96.
- Step 2: r³ = 8, so r = 2.
- Step 3: Using g₂: gₙ = 12 · 2ⁿ⁻². Equivalently, g₀ = 12/2² = 3, so gₙ = 3 · 2ⁿ.
- Step 4: g₈ = 3 · 2⁸ = 3 · 256 = 768.
Answer: gₙ = 12 · 2ⁿ⁻² = 3 · 2ⁿ; g₈ = 768.
- Example 3
Trap: where the counting starts
A sequence starts with a₁ = 7 and has common difference 4. Find a₁₀.
Show the solutionHide the solution
- Step 1: The first term is a₁, not a₀. From a₁ to a₁₀ is 9 steps, not 10.
- Step 2: Use aₙ = aₖ + d(n − k) with k = 1: a₁₀ = 7 + 4(10 − 1) = 7 + 36 = 43.
- Step 3: The trap answer is 7 + 4(10) = 47, which treats 7 as a₀.
Answer: a₁₀ = 43
Common mistakes
- Counting the steps between terms wrong. From aₖ to aₙ there are n − k steps.
- Using aₙ = a₀ + dn when the problem gives a₁ as the first term. Use aₙ = a₁ + d(n − 1) instead.
- Dividing instead of taking a root for a geometric ratio. If g₅ = g₂ · r³, then r is the cube root of g₅/g₂, not (g₅/g₂)/3.
On the exam
- Expect questions that give two terms and ask for the formula, or that ask which type of sequence fits a table or description.
- Sequences lead directly to topic 2.2. Notice that the arithmetic formula looks like a linear function and the geometric formula looks like an exponential one.
Connected topics
Videos
Check yourself
4 questions on 2.1 Change in Arithmetic and Geometric Sequences. Pick an answer to see if you got it, and why.
In an arithmetic sequence, a₃ = 11 and a₇ = 27. What is the value of a₂₀?
A geometric sequence has g₂ = 12 and g₅ = 96. What is the value of g₉?
The arithmetic sequence a has a₁ = 100 and common difference 50. The geometric sequence g has g₁ = 2 and common ratio 3. What is the least value of n for which gₙ > aₙ?
The first three terms of a geometric sequence are g₁ = 80, g₂ = 20 and g₃ = 5. Which of the following gives gₙ for n ≥ 1?
0 of 4 answered