Sequence Blueprint Study Guide
AP Precalculus Unit 2 • Topic 2.1 Discrete Functions
Master Reference Sheet
Sequence Blueprint Reference
Core formulas, discrete models, rate-of-change criteria, and series summation for the AP Exam.
Exam Quick-Sheet
Arithmetic Sequence Linear Model
Explicit Formula (1-based / 0-based):
\(a_n = a_1 + (n-1)d \quad \text{or} \quad a_n = a_0 + nd\)
Recursive Definition:
\(a_1 = \text{given}, \quad a_n = a_{n-1} + d \quad (n \ge 2)\)
Common Diff \(d\) \(d = a_n - a_{n-1}\)
Rate of Change \(\Delta a_n = d\) (Constant)
Function Equivalence: Graph consists of discrete collinear points on \(f(x) = dx + (a_1 - d)\) with domain \(n \in \mathbb{Z}^+\).
Geometric Sequence Exponential Model
Explicit Formula (1-based / 0-based):
\(g_n = g_1 \cdot r^{n-1} \quad \text{or} \quad g_n = g_0 \cdot r^n\)
Recursive Definition:
\(g_1 = \text{given}, \quad g_n = r \cdot g_{n-1} \quad (n \ge 2)\)
Common Ratio \(r\) \(r = \frac{g_n}{g_{n-1}}\)
Rate of Change \(\frac{g_{n+1}}{g_n} = r\) (Constant)
Function Equivalence: Graph consists of points on \(g(x) = \left(\frac{g_1}{r}\right)r^x = g_0 r^x\) with discrete domain \(n \in \mathbb{Z}^+\).
Series & Partial Sum Formulations
Finite Arithmetic Sum
\(S_n = \frac{n(a_1 + a_n)}{2}\)
Alternative: \(S_n = \frac{n}{2}[2a_1 + (n-1)d]\). Intuition: \(\text{number of terms} \times \text{average of endpoints}\).
Finite Geometric Sum
\(S_n = g_1 \frac{1 - r^n}{1 - r} \quad (r \ne 1)\)
Valid for any finite \(n\). If \(r = 1\), the sequence is constant and \(S_n = n \cdot g_1\).
Infinite Geometric Sum
\(S_\infty = \frac{g_1}{1 - r} \quad (|r| < 1)\)
Convergence: Sum exists if and only if \(|r| < 1\). If \(|r| \ge 1\), the infinite series diverges.
Decision Guide: How to Classify Given Data on the AP Exam
1. Table with Equal Steps
Calculate first differences: \(\Delta y = y_{k+1} - y_k\). If constant \(\implies\) Arithmetic. If ratios \(\frac{y_{k+1}}{y_k}\) are constant \(\implies\) Geometric.
2. Real-World Context Wording
Added/subtracted fixed amount each period \(\implies\) Arithmetic. Growing/decaying by percentage or factor (\(1 \pm r\)) \(\implies\) Geometric.
3. Logarithmic Linearity
If \(\log(y)\) vs \(x\) forms a straight line, \(y\) is Geometric with ratio \(r = 10^m\) (or \(e^m\) for natural log).
AP Precalculus Exam Traps & Notation Pitfalls
- • Term Counting in Sigma: \(\sum_{k=m}^p c_k\) has \((p - m + 1)\) terms. \(\sum_{k=3}^{11}\) has \(11 - 3 + 1 = 9\) terms, not 8.
- • Index Shift (\(n=0\) vs \(n=1\)): At initial time \(t=0\), use \(a_t = a_0 + dt\) and \(g_t = g_0 r^t\). At period 1, use \((n-1)\).
- • Negative Ratios (\(r < 0\)): Terms oscillate signs \((+,-,+,-)\). Even exponents yield positive; odd exponents yield negative.
- • Infinite Sum Criterion: Do not compute \(\frac{a_1}{1-r}\) unless you verify \(|r| < 1\). If \(|r| \ge 1\), answer "Diverges".
AP Precalculus • Study Guideline Page 1 of 5 • Reference Sheet
Concept Deep Dive Part 1 • Linear Discrete Models
Arithmetic Sequences & Series
Arithmetic Sequences as Discrete Linear Functions
Understanding constant first differences, non-consecutive term deduction, and partial sum applications.
The AP Precalc Definition of Arithmetic Behavior
A sequence is arithmetic if and only if the difference between successive terms is constant: \(\Delta a_n = a_{n+1} - a_n = d\) for all integers \(n \ge 1\). In AP Precalculus, this is recognized as a linear function evaluated at integer inputs with constant rate of change \(\frac{\Delta y}{\Delta x} = d\).
Technique: Finding Rule from Non-Consecutive Terms Standard AP Skill
Problem: An arithmetic sequence has \(a_5 = 19\) and \(a_{11} = 43\). Find the common difference \(d\) and write the explicit formula for \(a_n\).
Step 1: Find Slope \(d\) \(d = \frac{a_{11} - a_5}{11 - 5} = \frac{43 - 19}{6} = \frac{24}{6} = 4\)
Step 2: Solve for \(a_1\) \(a_5 = a_1 + 4d\)
\(19 = a_1 + 4(4) \implies a_1 = 3\)
Step 3: Explicit Formula \(a_n = 3 + 4(n - 1)\)
\(a_n = 4n - 1\)
Worked Example 1: Theater Seating Partial Sum Real-World Series
An outdoor concert venue has 22 seats in Row 1. Each consecutive row contains 3 additional seats. There are 28 rows in total. How many seats are in the 28th row, and what is the total venue capacity?
(A) Row 28 Count: \(a_{28} = a_1 + (28-1)d = 22 + 27(3) = 22 + 81 = \mathbf{103 \text{ seats}}\)
(B) Total Capacity: \(S_{28} = \frac{n(a_1 + a_{28})}{2} = \frac{28(22 + 103)}{2} = 14 \times 125 = \mathbf{1,750 \text{ seats}}\)
Verification using formula \(S_n = \frac{28}{2}[2(22) + 27(3)] = 14[44 + 81] = 1,750\).
Worked Example 2: Arithmetic Means Interpolation Algebraic Analysis
Insert 3 arithmetic means between 11 and 35. That is, find values \(m_1, m_2, m_3\) such that \(11, m_1, m_2, m_3, 35\) forms an arithmetic progression.
Let \(a_1 = 11\) and \(a_5 = 35\). Number of steps: \(5 - 1 = 4\).
\(d = \frac{a_5 - a_1}{5 - 1} = \frac{35 - 11}{4} = \frac{24}{4} = 6\)
The three means are: \(m_1 = 11 + 6 = \mathbf{17}\), \(m_2 = 17 + 6 = \mathbf{23}\), \(m_3 = 23 + 6 = \mathbf{29}\). Sequence: \(\{11, 17, 23, 29, 35\}\).
AP Free Response Justification Template:
When asked to justify why a model is arithmetic: "The sequence is arithmetic because the first differences between consecutive output values \(\Delta a_n = a_{n+1} - a_n = d\) are constant for equal increments of \(n\), indicating a constant rate of change characteristic of a linear function."
AP Precalculus • Study Guideline Page 2 of 5 • Arithmetic Sequences
Concept Deep Dive Part 2 • Exponential Discrete Models
Geometric Sequences & Series
Geometric Sequences as Discrete Exponential Functions
Multiplicative rate of change, infinite convergence limits, and semi-log transform behavior.
The AP Precalc Definition of Geometric Behavior
A sequence is geometric if and only if consecutive terms have a constant ratio: \(\frac{g_{n+1}}{g_n} = r\) for all integers \(n \ge 1\) (\(r \ne 0\)). In AP Precalculus, this represents an exponential function evaluated at discrete inputs: output values change by a constant multiplicative factor over equal input intervals.
High-Yield AP Connection: The Semi-Log Plot Essential Topic
Taking the logarithm of a geometric sequence produces an arithmetic sequence!
\(\ln(g_n) = \ln(g_1 \cdot r^{n-1}) = \ln(g_1) + (n - 1)\ln(r) = [\ln(r)]n + [\ln(g_1) - \ln(r)]\)
Exam Takeaway: If the plot of \(\ln(y)\) versus \(n\) is a straight line, the data is strictly geometric (exponential). The slope of that line is \(\ln(r)\) and the vertical intercept is \(\ln(g_0)\).
Worked Example 1: Rebounding Ball Total Distance Infinite Series
A rubber ball is dropped from a height of \(36\text{ ft}\). On each bounce, it rebounds to \(\frac{2}{3}\) of its previous height. Determine: (A) Height on the 4th bounce; (B) Total vertical distance traveled until it comes to rest.
(A) 4th Bounce Height: \(h_4 = 36 \left(\frac{2}{3}\right)^4 = 36 \cdot \frac{16}{81} = \frac{4 \cdot 16}{9} = \mathbf{\frac{64}{9}\text{ ft} \approx 7.11\text{ ft}}\)
(B) Total Vertical Distance:
The ball falls 36 ft once, then goes UP and DOWN for every rebound:
\(\text{Total} = 36 + 2 \times \sum_{k=1}^\infty 36\left(\frac{2}{3}\right)^k = 36 + 2 \times \frac{36(2/3)}{1 - 2/3} = 36 + 2 \times \frac{24}{1/3} = 36 + 2(72) = \mathbf{180\text{ ft}}\)
Worked Example 2: Determining Ratio with Odd Exponent Alternating Sign
In a geometric sequence, \(g_3 = -12\) and \(g_6 = 96\). Find \(r\), \(g_1\), and evaluate \(S_5\).
\(\frac{g_6}{g_3} = \frac{g_1 r^5}{g_1 r^2} = r^3 \implies r^3 = \frac{96}{-12} = -8 \implies r = \sqrt[3]{-8} = -2\)
\(g_3 = g_1 r^2 \implies -12 = g_1 (-2)^2 = 4 g_1 \implies g_1 = -3\)
Sum of first 5 terms: \(S_5 = (-3)\frac{1 - (-2)^5}{1 - (-2)} = (-3)\frac{1 - (-32)}{3} = (-1)(33) = \mathbf{-33}\).
AP Free Response Justification Template:
When asked to justify why a model is geometric: "The sequence is geometric because the ratio of consecutive output values \(\frac{g_{n+1}}{g_n} = r\) is constant for equal intervals of \(n\), which is the defining characteristic of an exponential relationship."
AP Precalculus • Study Guideline Page 3 of 5 • Geometric Sequences
AP Exam Prep Section 3 • Practice Problems
Targeted AP-Style Items
AP Precalculus Exam-Style Practice
Test analytical, graphical, and computational mastery under authentic AP exam constraints.
Question 1 • Tabular Rate Analysis (Multiple Choice) Calculator Inactive
Selected values of discrete function \(f\) on positive integers \(n\) are shown:
| \(n\) |
1 |
2 |
3 |
4 |
| \(f(n)\) |
16 |
24 |
36 |
54 |
Which statement correctly classifies \(f\) and gives its explicit formula for \(n \ge 1\)?
(A) Arithmetic, \(\Delta f = 8\); \(f(n) = 16 + 8(n-1)\)
(B) Geometric, \(\frac{f(n+1)}{f(n)} = 1.5\); \(f(n) = 16(1.5)^{n-1}\)
(C) Geometric, \(\frac{f(n+1)}{f(n)} = 1.5\); \(f(n) = 24(1.5)^n\)
(D) Neither arithmetic nor geometric model.
Question 2 • Semi-Log Linearization (Multiple Choice) Calculator Active
A sequence of positive terms \(y_n\) is plotted where the vertical axis is \(z = \ln(y_n)\). The points lie along a line through \((1, 4.2)\) with slope \(m = -0.35\). Which expression represents \(y_n\)?
(A) \(y_n = 4.2 - 0.35(n-1)\)
(B) \(y_n = e^{4.2} \cdot e^{-0.35(n-1)}\)
(C) \(y_n = e^{4.2} \cdot (-0.35)^{n-1}\)
(D) \(y_n = 4.2 \cdot (e^{-0.35})^{n-1}\)
Question 3 • Sigma Notation Evaluation (Multiple Choice) Calculator Inactive
Evaluate the finite arithmetic series: \(\sum_{k=4}^{20} (5k - 2)\).
(A) 986
(B) 918
(C) 988
(D) 1,020
Question 4 • AP Free Response Synthesis (Multi-Part) Full FRQ Item
An engineering firm monitors pollutant \(P(n)\), in \(\text{mg/L}\), in an industrial tank after \(n\) cycles of filtration. Initially (\(n=0\)), \(P(0) = 250\text{ mg/L}\). Each cycle removes \(18\%\) of the pollutant remaining from the preceding cycle.
Part (a): State whether \(P(n)\) represents an arithmetic or geometric sequence. Write an explicit expression for \(P(n)\) for \(n \ge 0\).
Part (b): Calculate the pollutant concentration remaining after \(4\) filtration cycles to the nearest hundredth.
Part (c): Evaluate \(\lim_{n \to \infty} P(n)\). Use sequence common ratio properties to interpret what this means in context.
AP Precalculus Exam Strategy Tips
Show Setup First: On calculator-active FRQs, always write down your unsimplified formula with substituted values before writing the calculator output.
State Domain Intervals: When justifying linear or exponential behavior, specify that the condition holds "over equal increments of input \(\Delta n\)".
Check Oscillations: If \(r < 0\), test \(n=1\) and \(n=2\) to ensure your formula's initial signs match the prompt's signs.
AP Precalculus • Study Guideline Page 4 of 5 • Exam Practice
Scoring Guide Section 4 • Detailed Solutions
AP Rationales & Rubric
Practice Problem Solutions & Scoring Rubric
Complete analytical derivations, distractor rationales, and AP Precalculus grading rubrics.
Question 1 Solution: Correct Answer is (B) Topic 2.1
Derivation: Check successive ratios: \(\frac{24}{16} = 1.5\), \(\frac{36}{24} = 1.5\), \(\frac{54}{36} = 1.5\). Since the ratio is constant, \(f\) is geometric with \(r = 1.5\) and initial term \(f(1) = 16\). Using \(f(n) = f(1)r^{n-1}\), we obtain \(f(n) = 16(1.5)^{n-1}\).
Distractor Note: Option (A) falsely assumes constant difference (\(24-16=8\), but \(36-24=12\)). Option (C) incorrectly uses 0-based exponent \(n\) with the 1-based initial term.
Question 2 Solution: Correct Answer is (B) Semi-Log Models
Derivation: In point-slope form: \(z - z_1 = m(n - 1) \implies \ln(y_n) - 4.2 = -0.35(n - 1) \implies \ln(y_n) = 4.2 - 0.35(n - 1)\). Exponentiating both sides yields \(y_n = e^{4.2 - 0.35(n-1)} = e^{4.2} \cdot e^{-0.35(n-1)} = e^{4.2}(e^{-0.35})^{n-1}\).
Distractor Note: Option (A) forgets to invert the natural log. Option (D) forgets that the initial coefficient must be \(e^{4.2} \approx 66.69\), not 4.2.
Question 3 Solution: Correct Answer is (A) Sigma Series
Derivation: Number of terms \(N = 20 - 4 + 1 = 17\). First term (\(k=4\)): \(a_4 = 5(4) - 2 = 18\). Last term (\(k=20\)): \(a_{20} = 5(20) - 2 = 98\). Sum: \(S = \frac{N(a_4 + a_{20})}{2} = \frac{17(18 + 98)}{2} = \frac{17(116)}{2} = 17 \times 58 = \mathbf{986}\).
Distractor Note: Option (C) incorrectly uses \(20 - 4 = 16\) terms (\(16 \times 58 = 928\)). Watch term counts!
Question 4 • AP Scoring Rubric & Exemplar Response 3 Points Total
Part (a) • 1 Point 1 pt: Correct Model & Equation
Exemplar: If \(18\%\) is removed, \(100\% - 18\% = 82\%\) remains. The common ratio is \(r = 0.82\). Because the remaining amount multiplies by a constant factor of \(0.82\) per filtration cycle, \(P(n)\) is a geometric sequence. With \(P(0) = 250\), the explicit formula is: \(P(n) = 250(0.82)^n \quad \text{for } n \ge 0\)
Part (b) • 1 Point 1 pt: Accurate Evaluation with Units
Exemplar: Evaluate at \(n = 4\): \(P(4) = 250(0.82)^4 = 250(0.45212176) \approx \mathbf{113.03\text{ mg/L}}\). After 4 filtration cycles, approximately \(113.03\text{ mg/L}\) of pollutant remains in the tank.
Part (c) • 1 Point 1 pt: Limit Justification via \(|r| < 1\)
Exemplar: As \(n \to \infty\), \(\lim_{n \to \infty} P(n) = \lim_{n \to \infty} 250(0.82)^n = 0\). Because \(P(n)\) is geometric with common ratio \(|r| = 0.82 < 1\), successive output values decrease toward zero, indicating that with unlimited cycles, all detectable pollutant will eventually be removed.
AP Precalc Exam Day Checklist: 1. Confirm \(n=0\) vs \(n=1\) • 2. Check \(\Delta y\) vs \(\frac{y_{n+1}}{y_n}\) • 3. Count terms \((p-m+1)\) • 4. Test \(|r| < 1\) for infinite sums
AP Precalculus • Study Guideline Page 5 of 5 • Solutions & Rubric
