Sequences Formulas

Why This Matters

Sequences and series are the backbone of pattern recognition in mathematics—and they're everywhere on your Pre-Calc exams. Whether you're calculating compound interest, analyzing population growth, or preparing for calculus concepts like limits and convergence, these formulas give you the tools to predict the $n$th term or find the sum of hundreds of terms in seconds. You're being tested on your ability to identify sequence types, apply the correct formula, and understand when series converge or diverge.

The key insight? Every sequence formula encodes a specific type of growth pattern. Arithmetic sequences grow by constant addition, geometric sequences grow by constant multiplication, and recursive sequences build each term from previous ones. Don't just memorize formulas—know what type of pattern each formula captures and when to reach for it. Master the connections between these formulas, and you'll handle any sequence problem thrown at you.

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Defining Individual Terms

Before you can sum a sequence, you need to find specific terms. These formulas let you calculate any term directly or build terms step-by-step.

Arithmetic Sequence Formula

• $a_n = a_1 + (n-1)d$—calculates the $n$th term using the first term $a_1$ and common difference $d$
• Linear growth pattern—each term increases (or decreases) by the same constant amount, creating an additive structure
• Identify by subtraction—if $a_{n} - a_{n-1}$ gives the same value for all consecutive pairs, you have an arithmetic sequence

Geometric Sequence Formula

• $a_n = a_1 \cdot r^{(n-1)}$—calculates the $n$th term using the first term $a_1$ and common ratio $r$
• Exponential growth pattern—each term is multiplied by the same factor, creating a multiplicative structure
• Identify by division—if $\frac{a_n}{a_{n-1}}$ gives the same value for all consecutive pairs, you have a geometric sequence

Explicit Sequence Formula

• $a_n = f(n)$—provides a direct calculation for any term using only the term number $n$
• No previous terms needed—jump straight to the 100th term without calculating terms 1-99
• Most efficient for finding specific terms—convert recursive definitions to explicit form when possible for faster computation

Recursive Sequence Formula

• $a_n = f(a_{n-1}, a_{n-2}, \ldots)$—defines each term based on one or more previous terms plus initial conditions
• Requires starting values—you must know $a_1$ (and sometimes $a_2$) to generate the sequence
• Natural for pattern description—some sequences like Fibonacci are most intuitively defined recursively, even if explicit formulas exist

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Summing Finite Sequences

When you need the total of multiple terms, series formulas save you from adding everything manually. Each formula matches its sequence type.

Arithmetic Series Sum Formula

• $S_n = \frac{n}{2}(a_1 + a_n)$—sums $n$ terms by averaging the first and last terms, then multiplying by the count
• Alternative form: $S_n = \frac{n}{2}(2a_1 + (n-1)d)$—useful when you know $d$ but haven't calculated $a_n$
• Gauss's insight—pairing terms from opposite ends gives equal sums, making this formula elegant and fast

Geometric Series Sum Formula

• $S_n = a_1 \cdot \frac{1 - r^n}{1 - r}$ for $r \neq 1$—sums $n$ terms of a geometric sequence
• Derived from factoring—multiplying the series by $r$ and subtracting creates telescoping cancellation
• Watch for $r = 1$—this special case means all terms equal $a_1$, so $S_n = n \cdot a_1$

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Infinite Series and Convergence

Infinite series introduce the critical concept of convergence—does adding forever give a finite answer?

Infinite Geometric Series Sum Formula

• $S = \frac{a_1}{1 - r}$ when $|r| < 1$—the series converges to this finite value
• Convergence condition is critical—if $|r| \geq 1$, the series diverges and has no finite sum
• Foundation for calculus limits—this formula demonstrates how infinite processes can yield finite results, a key concept you'll revisit

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Notation and Special Sequences

These tools help you communicate about sequences efficiently and recognize important patterns.

Sigma Notation

• $\sum_{i=m}^{n} f(i)$—compact notation where $i$ is the index, $m$ is the start, $n$ is the end, and $f(i)$ defines each term
• Essential for series work—translates "add up all terms from $m$ to $n$" into a single expression
• Properties simplify calculation—you can factor out constants, split sums, and shift indices using algebraic rules

Fibonacci Sequence Formula

• Recursive definition: $F_n = F_{n-1} + F_{n-2}$ with $F_0 = 0$ and $F_1 = 1$—each term sums the two before it
• Sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21...—memorize the first several terms for quick recognition
• Appears in nature and algorithms—from spiral shells to computer science efficiency analysis, this sequence has remarkable real-world applications

Arithmetic-Geometric Sequence Formula

• $a_n = (a_1 + (n-1)d) \cdot r^{(n-1)}$—combines linear growth with exponential growth in each term
• Hybrid structure—the arithmetic component $(a_1 + (n-1)d)$ gets multiplied by the geometric component $r^{(n-1)}$
• Financial applications—models scenarios like increasing payments with interest, common in annuity calculations

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Quick Reference Table

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Self-Check Questions

1. What's the key difference between identifying an arithmetic sequence versus a geometric sequence? Which operation do you perform on consecutive terms?

2. You're given a geometric series and asked to find its sum. What must you check before using the infinite series formula $S = \frac{a_1}{1-r}$?

3. Compare and contrast explicit and recursive formulas: when would you prefer each, and what information do you need to use them?

4. If an FRQ describes a sequence where each term is the sum of the two preceding terms, which formula type applies? What initial conditions must be given?

5. A problem gives you $a_1 = 5$, $a_{10} = 50$, and tells you the sequence is arithmetic. Which sum formula would you use, and why might you choose one version over the other?