Ask: term, position or total?
The requested quantity selects the mathematics: one value uses a_n, a count solves for n, and an accumulated total uses S_n.
Learning goals
- Model repeated equal change as an AP.
- Choose nth-term or sum reasoning from the requested quantity.
- Translate divisibility, rows, digits and geometry into AP parameters and validate results.
Concept lesson
First, last and count
For an interval, identify the first allowed term, last allowed term and common difference before finding n. Respect 'between' and endpoint conditions.
Select the AP tool
Use a_n for a particular value or boundary; solve a+(n-1)d=l for a count or position; use S_n for a combined total. Multi-step questions may require more than one tool.
Rows of rose plants
Rows contain 23,21,19,...,5 plants. Here a=23,d=-2,l=5. Solve 5=23+(n-1)(-2), giving n=10 rows.
Multiples inside an open interval
Between 201 and 401, the first multiple of 6 is 204 and last is 396. With d=6, solve 396=204+(n-1)6 to obtain n=33.
Validate the requested quantity
For each scenario, name whether the answer is a term, index, count or sum; verify that n is a positive integer and attach the correct unit.
Audit interval endpoints
For open and closed intervals, explicitly test each boundary for divisibility before constructing the AP and check the count by listing a small analogous case.
Recover a finite AP and verify a sum identity
Given n=16, S16=768 and l=93, use S_n=n(a+l)/2 to find a, then use l=a+15d to find d. State the AP and verify the required comparison of sums.
Extended AP answer structure
Define the AP quantities, translate all conditions, select formulas, solve, validate integer indices and contextual constraints, and state the requested result with units.
Exit ticket
Find the number and sum of integers strictly between 100 and 400 that leave remainder 1 when divided by 5. Identify endpoints, model the AP, choose formulas and validate the count.