A reliable method
Identify how adjacent terms change before choosing a formula. For a binomial coefficient, distinguish ordered selections from unordered groups.
The common trap
The nth arithmetic term uses n − 1 steps after the first term. Using n steps shifts the entire answer by one common difference.
Check and transfer
Cancel factorial products before calculating large numbers. For example, n!/(n − 2)! keeps only the final two factors, provided the integer domain is valid.
Worked comparison: identify the structure
An arithmetic sequence starting at 3 with difference 4 reaches term five after four additions: 3+4×4=19. A geometric sequence with first term 3 and ratio 4 reaches term five as 3×4⁴=768. Decide whether the step adds or multiplies.
Transfer and explain
The coefficient of x² in (1+x)⁵ is 5×4/2=10 because the order of the chosen two factors does not matter. Similarly, 7!/5!=7×6=42 after cancelling the full common factorial.
An arithmetic sequence starts at 4 and has common difference 4. What is term 6?
- Count one fewer step than the term number.
- Add that many common differences to the first term.
- The requested value is 24. Substitute or reverse the operation to check this result.
Compare the other answer choices
Count the changes but omit the first term.
Use n differences instead of n minus one.
Treat the sequence as geometric with ratio four.
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.
CLEP College Algebra exam overviewSiyavula Grade 10 foundation reference