AP Precalculus
AP Precalc 2.1–2.3: FRQ Practice
Shared by Ashton · Oct 25, 2026 · 0 plays
5 multi-part free-response questions for 2.1–2.3 (two sequences, which kind of function, reading an exponential graph, modeling with sequences, find the error), split into 23 parts. Each part has a rubric and a model answer written the way you'd write it on the test.
23 questions across 5 sections. Write your answer, then check it against the rubric and score yourself.
Sample questions
Open an item to see its answer.
The first six terms of an arithmetic sequence aₙ and a geometric sequence gₙ are graphed as separate points. All terms of gₙ are positive. You know that a₂ = g₂ = 4, a₆ = 16 and g₆ = 64. No calculator. — (a) Find the common difference of aₙ and the common ratio of gₙ. Then write an expression for each. Show the work that leads to your answers.
Sample answer: d = (16 − 4)/(6 − 2) = 12/4 = 3, so aₙ = 4 + 3(n − 2), which simplifies to 3n − 2. For gₙ: 64 = 4r⁴, so r⁴ = 16 and r = ±2. The terms are positive, so r = 2 and gₙ = 4(2)^(n − 2), which simplifies to 2ⁿ.
The first six terms of an arithmetic sequence aₙ and a geometric sequence gₙ are graphed as separate points. All terms of gₙ are positive. You know that a₂ = g₂ = 4, a₆ = 16 and g₆ = 64. No calculator. — (b) Find the least value of n for which gₙ > aₙ + 100. Show the values you compared.
Sample answer: At n = 6: g₆ = 64 and a₆ + 100 = 116, so 64 < 116. At n = 7: g₇ = 128 and a₇ + 100 = 119, so 128 > 119. The least value is n = 7.
The first six terms of an arithmetic sequence aₙ and a geometric sequence gₙ are graphed as separate points. All terms of gₙ are positive. You know that a₂ = g₂ = 4, a₆ = 16 and g₆ = 64. No calculator. — (c) The linear function f(x) = 3x − 2 has the same rule as aₙ. Explain how the domain of f differs from the domain of aₙ. Use that to explain why the graph of aₙ is a set of points and not a line.
Sample answer: The domain of f is all real numbers. The domain of aₙ is only whole numbers, n = 1, 2, 3, … The sequence has no values between whole numbers, such as n = 2.5, so its graph is separate points that lie on the line y = 3x − 2 but aren't connected.
The first six terms of an arithmetic sequence aₙ and a geometric sequence gₙ are graphed as separate points. All terms of gₙ are positive. You know that a₂ = g₂ = 4, a₆ = 16 and g₆ = 64. No calculator. — (d) Write limit statements for lim(n→∞) aₙ and lim(n→∞) gₙ. Challenge: a different geometric sequence qₙ also has q₂ = 4 and q₆ = 64, but its terms are NOT required to be positive. Find another possible common ratio, write qₙ, and describe how its terms behave as n increases.
Sample answer: lim(n→∞) aₙ = ∞ and lim(n→∞) gₙ = ∞. Challenge: r = −2 also satisfies r⁴ = 16, so qₙ = 4(−2)^(n − 2), giving 4, −8, 16, −32, 64, … The terms alternate in sign and grow in size without bound, so they don't approach any single value. The limit doesn't exist, and it isn't ∞ or −∞ either.
Values of three functions at selected inputs: x: 0, 2, 4, 6, 8 | f(x): 5, 11, 17, 23, 29 | g(x): 96, 24, 6, 1.5, 0.375 | h(x): 1, 3, 9, 19, 33 | No calculator. — (a) For each function, decide whether it is best modeled by a linear, quadratic or exponential function. Justify each answer using input and output values.
Sample answer: f is linear, because over equal-length input-value intervals of length 2, the output values change by the same amount, +6. g is exponential, because over equal-length input-value intervals of length 2, the output values are multiplied by the same factor, ¼, so the outputs are proportional. h is quadratic, because over equal-length input-value intervals, the first differences 2, 6, 10, 14 change by a constant amount, 4, so the second differences are constant.