Decision and conditions
Identify whether a recurrence, an alternating sign or a ratio controls the next term. The first term has index one. A recurrence acts on the previous term, not the index.
Worked model
A sequence has a₁ = 3, and aₙ₊₁ = 2aₙ + 1. What is a₄? a₂ = 7. a₃ = 15. a₄ = 31. Before calculating: Apply the recurrence three times after the first term.
Contrast: a different decision
An arithmetic sequence has third term 11 and seventh term 23. What is its first term? The difference across four steps is 12, so the common difference is 3. Move back two steps from the third term. The first term is 11 − 2(3) = 5. Before calculating: How many equal differences separate the third and seventh terms?
Transfer with solution
A geometric sequence has positive common ratio, second term 6, and fourth term 54. What is the third term? The squared ratio is 9, so the positive ratio is 3. Multiply the second term 6 by 3. The third term is 18. Before calculating: The ratio is applied twice between the two known terms.
Check the boundary
A visible difference in two terms does not justify assuming a constant difference everywhere. What is the sum 1 − 2 + 3 − 4 + … + 5 − 6? Each consecutive pair has sum −1. There are 3 pairs. The total is -3.
A sequence has a₁ = 3, and aₙ₊₁ = 2aₙ + 1. What is a₄?
- a₂ = 7.
- a₃ = 15.
- a₄ = 31.
Compare the other answer choices
Doubles the first term once and then adds one for each of three transitions.
Stops at the third term instead of the fourth.
Doubles three times but omits every added one.
Applies four transitions and reports the fifth term.
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.
GRE Quantitative Reasoning overview