Sequence Formulas

Why This Matters

Sequences and series are the backbone of pattern recognition in mathematics—and they show up everywhere in Honors Algebra II, from modeling real-world growth to laying the foundation for calculus. You're being tested on your ability to distinguish between arithmetic vs. geometric behavior, recursive vs. explicit representations, and finite vs. infinite sums. These aren't just formulas to memorize; they're tools for understanding how quantities grow, accumulate, and sometimes converge to predictable limits.

Here's the key insight: every sequence formula encodes a specific type of pattern. Arithmetic sequences grow by addition (linear growth), while geometric sequences grow by multiplication (exponential growth). When you understand why each formula works, you can derive them on the fly, recognize which to apply in word problems, and handle FRQ-style questions that ask you to compare or combine different sequence types. Don't just memorize formulas—know what mathematical behavior each one captures.

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Arithmetic Patterns: Growth by Addition

Arithmetic sequences and series are built on constant differences—each term increases (or decreases) by the same amount. This linear pattern produces predictable, steady growth that's easy to model and sum.

Arithmetic Sequence Formula

• $a_n = a_1 + (n-1)d$ gives you any term directly—$a_1$ is the first term, $d$ is the common difference
• Common difference $d$ stays constant throughout; find it by subtracting any term from the next: $d = a_{n+1} - a_n$
• Linear relationship—plotting term number vs. term value gives a straight line with slope $d$

Arithmetic Series Sum Formula

• $S_n = \frac{n}{2}(a_1 + a_n)$ calculates the sum of $n$ terms by averaging the first and last terms, then multiplying by $n$
• Alternate form $S_n = \frac{n}{2}(2a_1 + (n-1)d)$ works when you don't know the last term—use this when given only $a_1$, $d$, and $n$
• Gauss's insight—pairing terms from opposite ends gives equal sums, which is why the average-and-multiply approach works

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Geometric Patterns: Growth by Multiplication

Geometric sequences multiply by a constant ratio each time, producing exponential behavior. This multiplicative pattern models compound interest, population growth, and decay processes.

Geometric Sequence Formula

• $a_n = a_1 \cdot r^{n-1}$ gives any term directly—$a_1$ is the first term, $r$ is the common ratio
• Common ratio $r$ is found by dividing any term by the previous one: $r = \frac{a_{n+1}}{a_n}$
• Exponential behavior—when $|r| > 1$, terms explode; when $|r| < 1$, terms shrink toward zero

Geometric Series Sum Formula (Finite)

• $S_n = a_1 \cdot \frac{1 - r^n}{1 - r}$ sums the first $n$ terms—only valid when $r \neq 1$
• Derivation trick—multiply $S_n$ by $r$, subtract from original $S_n$, and most terms cancel (a classic exam derivation question)
• Watch the signs—if $r > 1$, you can rewrite as $S_n = a_1 \cdot \frac{r^n - 1}{r - 1}$ to avoid negative numerators

Infinite Geometric Series Sum Formula

• $S = \frac{a_1}{1 - r}$ gives the sum of infinitely many terms—but only when $|r| < 1$
• Convergence condition—the series converges because terms shrink to zero fast enough; if $|r| \geq 1$, the sum diverges (doesn't exist)
• Calculus preview—this formula introduces limits and is your first encounter with infinite processes having finite answers

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Explicit vs. Recursive: Two Ways to Define Sequences

Every sequence can be described two ways: explicitly (direct formula for any term) or recursively (each term depends on previous terms). Understanding both representations—and converting between them—is a core Algebra II skill.

Explicit Sequence Formula

• $a_n = f(n)$ computes any term directly from its position—no previous terms needed
• Efficiency advantage—want the 100th term? Plug in $n = 100$ without calculating terms 1–99
• Arithmetic and geometric formulas are explicit—$a_n = a_1 + (n-1)d$ and $a_n = a_1 \cdot r^{n-1}$ both fit this pattern

Recursive Sequence Formula

• $a_n = f(a_{n-1}, a_{n-2}, \ldots)$ defines each term using previous terms, plus initial conditions to start the sequence
• Intuitive for modeling—recursive definitions often match how real processes work (today's population depends on yesterday's)
• Requires iteration—finding the 100th term means calculating all 99 terms before it, unless you convert to explicit form

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Special Sequences: Fibonacci and Hybrid Patterns

Some sequences don't fit neatly into arithmetic or geometric categories—they combine patterns or follow unique rules. These special cases test your ability to apply sequence concepts flexibly.

Fibonacci Sequence Formula

• Recursive definition $F_n = F_{n-1} + F_{n-2}$ with $F_0 = 0$ and $F_1 = 1$—each term is the sum of the two before it
• Binet's explicit formula $F_n = \frac{\phi^n - (1-\phi)^n}{\sqrt{5}}$ where $\phi = \frac{1 + \sqrt{5}}{2}$ is the golden ratio—remarkably, this always produces integers
• Appears everywhere—from spiral patterns in nature to algorithm analysis; it's the classic example of recursive-to-explicit conversion

Arithmetic-Geometric Sequence Formula

• $a_n = (a_1 + (n-1)d) \cdot r^{n-1}$ combines linear and exponential growth—the arithmetic part gets multiplied by the geometric part
• Financial applications—models scenarios like increasing payments with compound interest
• Harder to sum—series formulas for these require techniques beyond basic Algebra II, but recognizing the pattern is testable

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Connections to Combinatorics: The Binomial Theorem

The Binomial Theorem generates sequences of coefficients that appear throughout algebra and probability. It's the bridge between sequences and polynomial expansion.

Binomial Theorem for Sequences

• $\displaystyle (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$ expands any binomial power into a sum of terms
• Binomial coefficients $\binom{n}{k} = \frac{n!}{k!(n-k)!}$ form Pascal's Triangle—each row is a sequence with its own patterns
• Connects to probability—the coefficients count combinations, making this essential for binomial distributions and counting problems

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Quick Reference Table

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Self-Check Questions

1. Both arithmetic and geometric sequences have explicit formulas for $a_n$. What's the key structural difference between $a_n = a_1 + (n-1)d$ and $a_n = a_1 \cdot r^{n-1}$, and what type of growth does each represent?

2. You're given a geometric series with $a_1 = 12$ and $r = 0.5$. Can you find both the sum of the first 6 terms AND the sum of infinitely many terms? Which formula applies to each?

3. Compare and contrast explicit and recursive formulas: If you need to find the 50th term of a sequence quickly, which type do you want? If you're modeling a process where each step depends on the previous one, which is more natural?

4. The Fibonacci sequence is defined recursively, but Binet's formula is explicit. Why might you prefer the recursive definition for understanding the pattern, but the explicit formula for computation?

5. An FRQ gives you a sequence: $3, 6, 12, 24, \ldots$ and asks for the sum of the first 10 terms. Identify the sequence type, write the appropriate sum formula, and explain why you can't use the infinite series formula here.