Knowledge Notes | Carey AP Precalculus Practice Lab

For consecutive sequence indices, check both the repeated change and an anchor term.

What a strong response shows

Check differences and ratios at consecutive indices.
Keep the starting index with its starting term.
A sequence is defined on whole-number indices, not a continuous interval.

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What You Need to Know

Key idea

A sequence is a function whose inputs are whole-number indices. Arithmetic terms change by the same amount; geometric terms change by the same factor.

Fast decision rule

Subtract consecutive terms first. If the differences match, use arithmetic; if the ratios match, use geometric.

Worked steps

For $12$ , $17$ , $22$ , the difference is $5$ each time. For $3$ , $6$ , $12$ , the ratio is $2$ each time.

Common mistake

Calling a sequence geometric because it increases, even when its ratios are not constant.

Mini check

Which test should you use for $8$ , $12$ , $16$ , $20$ ?

Learning Objectives

Key idea

2.1.A — Express arithmetic sequences found in mathematical and contextual scenarios as functions of the whole numbers. 2.1.B — Express geometric sequences found in mathematical and contextual scenarios as functions of the whole numbers.

Fast decision rule

Match the prompt verb to the representation it asks you to build or interpret.

Worked steps

$1$ . Name the given quantities. $2$ . Choose the matching function or representation. $3$ . Show the parameter or operation that changes. $4$ . Check one value in context.

Common mistake

Starting calculations before identifying what the input and output mean.

Mini check

Can you state what the input, output, and requested evidence are before choosing a formula?

Key Concepts

Key idea

The index $n$ identifies a term; the common difference $d$ controls additive change; the common ratio $r$ controls multiplicative change.

Fast decision rule

Use a difference for an arithmetic pattern and a ratio for a geometric pattern.

Worked steps

A sequence that starts at $10$ and adds $3$ has $d=3$ . A sequence that starts at $10$ and multiplies by $3$ has $r=3$ .

Common mistake

Using a common ratio when the pattern only has a constant difference.

Mini check

What stays constant in a geometric sequence?

Relevant Representations

Key idea

A sequence can be shown by a context, table, list, graph of discrete points, recursive rule, or explicit rule.

Fast decision rule

Use a table to test change, then use the explicit rule when the prompt asks for any term.

Choose the sequence representation
Representation	Best evidence	Check
List	first few terms	difference or ratio
Table	index-to-term pairs	units and index
Graph	discrete behavior	whole-number inputs
Rule	any term	initial value and change

Worked steps

Place the index in the first column and the term in the second. Check two neighboring rows before writing the rule.

Common mistake

Connecting sequence points with a continuous curve when the inputs are only whole numbers.

Mini check

Which representation makes the index restriction easiest to see?

Essential Relationships or Formulas

Key idea

Use an exact relationship to expose the pattern before substituting values.

a_n=a_1+(n-1)d

a_n=a_1r^{n-1}

Fast decision rule

Keep the variables and restrictions visible, then substitute with units attached.

Worked example

Write the relationship, identify each parameter, substitute the known values, and check one output.

Common mistake

Using a familiar formula without checking its domain, base, interval, or unit assumptions.

Mini check

Which symbol or parameter carries the feature the prompt asks you to find?

Worked Examples

Key idea

A short, labeled chain of steps is easier to check than a page of unexplained algebra.

Fast decision rule

Write the representation first, calculate second, and interpret last.

Worked steps

For an arithmetic sequence with first term $7$ and difference $4$ , use the explicit rule to find the tenth term. For a geometric sequence with first term $7$ and ratio $4$ , use the power rule instead.

Common mistake

Skipping the interpretation step and reporting an answer with no units, domain, or model evidence.

Mini check

Which step would you show so another student could reproduce the result?

Common Mistakes

Key idea

Most errors come from a mismatch between the pattern, the operation, and the allowed input.

Fast decision rule

Before submitting, check pattern, operation, domain, and units in that order.

Worked steps

Circle the first line where the pattern or restriction changes. Correct that line, then recompute only the affected steps.

Common mistake

Trusting a calculator output that hides a wrong model, sign, base, or domain.

Mini check

What independent check would catch your most likely error?

AP Exam Tips

Key idea

Use the topic workflow to connect the representation, the equation, and the context before calculating.

Fast decision rule

Read the prompt, choose the smallest reliable representation, and verify the result in context.

Worked steps

$1$ . Identify the pattern. $2$ . Write the matching form. $3$ . Show the key substitution or transformation. $4$ . Check units, domain, and direction.

Common mistake

Submitting a number without explaining why the selected model or transformation fits.

Mini check

Can you name the evidence that supports your final model or answer?

Practice

Open this topic’s practice set to apply the method. Work independently, then compare the evidence in your reasoning with the checked explanation.

TRY IT IN A NEW SETTING

Can you transfer the method?

Write an explicit rule for a geometric sequence with $a_3=20$ and ratio $1/2$ .

Compare your reasoning

An anchored rule is $a_{n}=20 \left(\frac{1}{2}\right)^{n-3}$ , for the sequence’s whole-number indices. At $n=3$ the exponent is zero, so the anchor is preserved.

Use it in a real question

Work independently, then compare the mathematical evidence in your answer.

Practise this topic Check the whole unit

2.1 · Change in Arithmetic and Geometric Sequences

Topic 2.1 · Change in Arithmetic and Geometric Sequences

What a strong response shows

What You Need to Know

Learning Objectives

Key Concepts

Relevant Representations

Essential Relationships or Formulas

Worked Examples

Common Mistakes

AP Exam Tips

Can you transfer the method?