Decision and conditions
Sum variable increments, track ordered updates and reconstruct endpoints from geometric constraints. A recurrence determines later terms from previous outputs; indices start at one.
Worked example
A sequence has a₁=1 and aₙ=aₙ₋₁+(2n−1) for every integer n≥2. What is a_8−a_3? The requested difference sums the five increments from index 4 through 8. Those increments are 7, 9, 11, 13, 15. Their sum is 55. Method choice: Subtracting the two terms cancels the earlier history; sum only the intervening increments.
Contrast: change the reasoning
Start with 3. Apply these four operations in order: add 3, multiply by 2, add 3, multiply by 2. What is the final value? After adding and doubling once, the value is 2(3+3)=12. Add three again to obtain 15. Double this result to obtain 30. Method choice: Track the output after each operation; later multiplication also acts on earlier additions.
Transfer with solution
A sequence repeats the three-term block 3, −3, 2, starting with 3. What is the sum of its first 10 terms? Each complete three-term block has sum 3−3+2=2. There are 3 complete blocks and one further term, which equals 3. The total is 2(3)+3=9. Method choice: Separate complete periods from the remaining prefix and sum both contributions.
Boundary check
Three positive numbers x<y<z form consecutive terms of a geometric sequence. Their sum is 21, and y=6. What is z−x? The geometric condition gives xz=y²=36; the sum gives x+z=15. (z−x)²=(x+z)²−4xz=225−144=81. The terms increase, so z−x is the positive root, 9. Method choice: Use the geometric middle term to obtain the product of the endpoints, then combine it with their sum.
Misconception to reject
A recurrence is not automatically an arithmetic or geometric sequence. Reports the number of intervening increments instead of adding their values.
A sequence has a₁=1 and aₙ=aₙ₋₁+(2n−1) for every integer n≥2. What is a_8−a_3?
- The requested difference sums the five increments from index 4 through 8.
- Those increments are 7, 9, 11, 13, 15.
- Their sum is 55.
Compare the other answer choices
Reports the number of intervening increments instead of adding their values.
Uses only the final increment.
Repeats the first increment five times although later increments grow.
Computes the later term but omits subtracting the earlier 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.
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