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.

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.

---

Arithmetic Patterns: Growth by Addition

Arithmetic sequences and series are built on constant differences. Each term increases (or decreases) by the same amount, producing a linear pattern that's straightforward 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: $d = a_{n+1} - a_n$. For example, in the sequence $5, 8, 11, 14, \ldots$, the common difference is $d = 8 - 5 = 3$.
• Linear relationship. If you plot term number vs. term value, you get a straight line with slope $d$ and y-intercept $a_1 - d$.

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 you're given only $a_1$, $d$, and $n$.
• Why it works (Gauss's insight): If you pair terms from opposite ends of the sequence, each pair has the same sum. For instance, in $1 + 2 + 3 + \ldots + 100$, pairing $1 + 100$, $2 + 99$, $3 + 98$, etc., gives 50 pairs that each sum to 101, so $S_{100} = 50 \times 101 = 5050$.

---

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: $r = \frac{a_{n+1}}{a_n}$. For example, in $2, 6, 18, 54, \ldots$, the common ratio is $r = \frac{6}{2} = 3$.
• Exponential behavior. When $|r| > 1$, terms grow without bound. When $|r| < 1$, terms shrink toward zero. When $r$ is negative, terms alternate in sign.

Geometric Series Sum Formula (Finite)

• $S_n = a_1 \cdot \frac{1 - r^n}{1 - r}$ sums the first $n$ terms. This is only valid when $r \neq 1$. (If $r = 1$, every term equals $a_1$, so $S_n = n \cdot a_1$.)
• Derivation trick. Multiply $S_n$ by $r$, then subtract from the original $S_n$. Most terms cancel, leaving $S_n - rS_n = a_1 - a_1 r^n$, which factors to $S_n(1 - r) = a_1(1 - r^n)$. This is a classic exam derivation question.
• Sign tip. If $r > 1$, you can rewrite as $S_n = a_1 \cdot \frac{r^n - 1}{r - 1}$ to avoid negative values in both the numerator and denominator.

Infinite Geometric Series Sum Formula

• $S = \frac{a_1}{1 - r}$ gives the sum of infinitely many terms, but only when $|r| < 1$.
• Why it converges. When $|r| < 1$, each successive term gets smaller and smaller, approaching zero. The partial sums settle toward a finite value. If $|r| \geq 1$, the terms don't shrink, so the sum diverges (no finite answer exists).
• Calculus preview. This formula is your first encounter with infinite processes producing finite answers, which is the core idea behind limits.

---

Explicit vs. Recursive: Two Ways to Define Sequences

Every sequence can be described two ways: explicitly (a 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 number. No previous terms needed.
• Efficiency advantage. Want the 100th term? Plug in $n = 100$ without calculating terms 1 through 99.
• The arithmetic and geometric formulas you already know are explicit: $a_n = a_1 + (n-1)d$ and $a_n = a_1 \cdot r^{n-1}$ both let you jump straight to any term.

Recursive Sequence Formula

• $a_n = f(a_{n-1}, a_{n-2}, \ldots)$ defines each term using previous terms, plus initial conditions (starting values) to anchor the sequence.
• Natural for modeling. Recursive definitions often match how real processes work. For example, a bank balance today equals yesterday's balance plus interest earned.
• Requires iteration. Finding the 100th term means calculating all 99 terms before it, unless you can convert to an explicit form.

---

Special Sequences: Fibonacci and Hybrid Patterns

Some sequences don't fit neatly into arithmetic or geometric categories. They combine patterns or follow unique rules, and they 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, giving $0, 1, 1, 2, 3, 5, 8, 13, 21, \ldots$
• Binet's explicit formula: $F_n = \frac{\phi^n - (1-\phi)^n}{\sqrt{5}}$ where $\phi = \frac{1 + \sqrt{5}}{2} \approx 1.618$ is the golden ratio. Despite the irrational numbers involved, this always produces integers.
• Why it matters. The Fibonacci sequence appears in spiral patterns in nature, algorithm analysis, and more. It's the classic example of converting a recursive definition into an explicit formula.

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. This models scenarios like making increasing payments into an account that earns compound interest.
• Harder to sum. Series formulas for these require techniques beyond standard Algebra II, but recognizing the pattern itself is testable.

---

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 itself a sequence with useful symmetry properties: $\binom{n}{k} = \binom{n}{n-k}$.
• Connects to probability. The coefficients count combinations, making this essential for binomial distributions and counting problems.

---

Quick Reference Table

---

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 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. A problem gives you the 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.