A reliable method
State whether indexing begins at zero or one. Write several terms and verify your formula at the first listed index before evaluating a later term.
The common trap
Using n instead of n-1 in a formula with a first term at n=1 shifts every prediction. A sequence rule does not automatically apply to noninteger inputs.
Make the connection
Arithmetic sequences connect to linear models; geometric sequences connect to exponential models. Their partial sums describe accumulated quantities.
Indexing and accumulation are separate
An arithmetic sequence starts a₁=3 with difference 4, so a₅=3+4(5−1)=19. A geometric sequence starts b₁=3 with ratio 2, so b₅=3×2⁴=48. In both cases the index 5 requires four transitions from term 1. The sum of the first four geometric terms is 3+6+12+24=45, not the fourth term 24. State whether the question requests one term or an accumulated total. Writing a short list provides a strong check for an off-by-one error before using a general formula for a large index.
Transfer, explain and check
A geometric sequence begins 2,6,18. Find its fourth term and the sum of its first four terms. Check: 54 and 80. Explain why their values answer different questions.
An arithmetic sequence has first term 4 and common difference 3. What is its sixth term?
- From the first to sixth term there are five steps.
- Add the difference 3 five times.
- The term is 4+5(3)=19.
Compare the other answer choices
Drop the first term after counting intervals.
Count six increments after the first term.
Multiply the first term by the index and ignore the difference.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
AP Precalculus course and syllabus