Fundamental Sequence Properties

Why This Matters

Sequences are the backbone of precalculus and set you up for everything you'll encounter in calculus. When you see a sequence, you're really looking at a function whose domain is positive integers—and that shift in thinking unlocks how we analyze growth patterns, long-term behavior, and summation. The AP exam loves testing whether you can identify sequence types, write formulas, and predict what happens as terms continue indefinitely.

Here's the key: you're being tested on your ability to recognize structure in number patterns and translate that structure into mathematical language. Don't just memorize that arithmetic sequences add and geometric sequences multiply—know why each formula works, when to use recursive versus explicit forms, and how boundedness and monotonicity connect to convergence. Master these relationships, and sequence problems become predictable.

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Sequence Types and Their Defining Features

The first skill is recognizing what kind of sequence you're dealing with. Each type has a signature pattern—arithmetic sequences grow by addition, geometric sequences grow by multiplication, and recursive sequences build from previous terms.

Arithmetic Sequences

• Constant common difference $d$—every term is found by adding the same value to the previous term
• Explicit formula $a_n = a_1 + (n-1)d$ allows direct calculation of any term without computing all previous terms
• Linear growth pattern means graphing $(n, a_n)$ produces points along a straight line

Geometric Sequences

• Constant common ratio $r$—every term is found by multiplying the previous term by the same value
• Explicit formula $a_n = a_1 \cdot r^{(n-1)}$ shows the exponential relationship between position and term value
• Exponential behavior means these sequences grow (or decay) much faster than arithmetic sequences when $|r| > 1$ (or $|r| < 1$)

Fibonacci Sequence

• Recursive definition $F_n = F_{n-1} + F_{n-2}$ with base cases $F_0 = 0$ and $F_1 = 1$—each term depends on the two preceding terms
• Golden ratio connection—the ratio $\frac{F_n}{F_{n-1}}$ approaches $\phi \approx 1.618$ as $n$ increases
• Natural applications appear in plant growth, shell spirals, and algorithm design, making this a favorite for conceptual questions

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Formulas: Recursive vs. Explicit

Understanding how to express a sequence mathematically is just as important as recognizing its type. Recursive formulas show the building process; explicit formulas give you direct access to any term.

Definition of a Sequence

• Ordered list of terms where each term $a_n$ corresponds to a specific position $n$ (a positive integer)
• Domain restriction—unlike continuous functions, sequences only exist at integer inputs, creating discrete points
• Finite or infinite—sequences can terminate after a set number of terms or continue indefinitely

Recursive vs. Explicit Formulas

• Recursive formulas define $a_n$ in terms of previous terms—e.g., $a_n = a_{n-1} + d$ requires knowing $a_{n-1}$ first
• Explicit formulas calculate $a_n$ directly from $n$—faster for finding the 100th term without computing terms 1-99
• Conversion skill is frequently tested: given one form, can you derive the other?

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Long-Term Behavior: Convergence and Limits

The AP exam frequently asks what happens to a sequence "in the long run." Convergence describes whether terms settle toward a specific value; divergence means they don't.

Convergence and Divergence

• Convergent sequences approach a specific finite value (the limit) as $n \to \infty$
• Divergent sequences either grow without bound, oscillate, or otherwise fail to approach a single value
• Geometric test—a geometric sequence converges only when $|r| < 1$, approaching zero

Limits of Sequences

• Limit notation $\lim_{n \to \infty} a_n = L$ means terms get arbitrarily close to $L$ for sufficiently large $n$
• Existence requirement—not all sequences have limits; only convergent sequences do
• Practical interpretation—the limit tells you the sequence's "destination" even if it never actually arrives

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Structural Properties: Boundedness and Monotonicity

These properties help you analyze sequences without computing every term. Bounded sequences stay within limits; monotonic sequences move consistently in one direction.

Bounded Sequences

• Upper bound exists if all terms satisfy $a_n \leq M$ for some real number $M$
• Lower bound exists if all terms satisfy $a_n \geq m$ for some real number $m$
• Convergence connection—bounded monotonic sequences are guaranteed to converge (a key theorem)

Monotonic Sequences

• Increasing sequences satisfy $a_n \leq a_{n+1}$ for all $n$—terms never decrease
• Decreasing sequences satisfy $a_n \geq a_{n+1}$ for all $n$—terms never increase
• Monotonic Convergence Theorem—if a sequence is both monotonic and bounded, it must converge

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Summation and Series Notation

When you add up sequence terms, you enter the world of series. Sigma notation provides a compact way to express sums, which is essential for series analysis.

Sigma Notation for Series

• Summation symbol $\sum$ compactly represents adding terms: $\sum_{i=1}^{n} a_i = a_1 + a_2 + \cdots + a_n$
• Index variable (often $i$, $k$, or $n$) indicates which terms are being summed and over what range
• Series convergence—when summing infinitely many terms, the sum may converge to a finite value or diverge

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Quick Reference Table

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Self-Check Questions

1. What do arithmetic and geometric sequences have in common structurally, and what is the key difference in how their terms change?

2. Given the recursive formula $a_n = a_{n-1} \cdot 3$ with $a_1 = 2$, write the explicit formula and determine whether the sequence converges or diverges.

3. Compare and contrast bounded sequences and monotonic sequences—can a sequence be one without being the other? Give examples.

4. A geometric sequence has $a_1 = 100$ and $r = 0.5$. What is $\lim_{n \to \infty} a_n$, and why does this sequence converge while one with $r = 2$ does not?

5. (FRQ-style) The Fibonacci sequence is defined recursively. Explain why it cannot be easily expressed with a simple explicit formula like arithmetic or geometric sequences, and describe one mathematical property that emerges from its recursive structure.