Sequences Formulas

Why This Matters

Sequences and series are the backbone of pattern recognition in mathematics, and they show up constantly on Pre-Calc exams. Whether you're calculating compound interest, analyzing population growth, or building toward calculus concepts like limits and convergence, these formulas let you predict the $n$th term or find the sum of hundreds of terms in seconds.

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 use it.

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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 the common difference $d$
• This is a linear growth pattern: each term increases (or decreases) by the same constant amount
• How to identify one: subtract consecutive terms. If $a_{n} - a_{n-1}$ gives the same value for every pair, it's arithmetic

Geometric Sequence Formula

• $a_n = a_1 \cdot r^{(n-1)}$ calculates the $n$th term using the first term $a_1$ and the common ratio $r$
• This is an exponential growth pattern: each term is multiplied by the same factor
• How to identify one: divide consecutive terms. If $\frac{a_n}{a_{n-1}}$ gives the same value for every pair, it's geometric

Explicit Sequence Formula

• $a_n = f(n)$ provides a direct calculation for any term using only the term number $n$
• No previous terms needed. You can jump straight to the 100th term without calculating terms 1 through 99.
• When possible, convert recursive definitions to explicit form 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
• You must know $a_1$ (and sometimes $a_2$) to generate the sequence
• Some sequences, like Fibonacci, are most naturally defined recursively, even though 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 by hand. 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: $S_n = \frac{n}{2}(2a_1 + (n-1)d)$, which is useful when you know $d$ but haven't calculated $a_n$ yet
• The logic behind this (Gauss's insight): pairing terms from opposite ends of the sequence always gives equal sums, so you only need one pair times the number of pairs

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
• This formula comes from multiplying the series by $r$ and subtracting, which causes most terms to cancel (telescoping)
• Watch for $r = 1$: that 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 terms forever actually 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.
• The convergence condition is non-negotiable. If $|r| \geq 1$, the series diverges and has no finite sum. Always check this before applying the formula.
• This formula demonstrates how an infinite process can yield a finite result, a concept that becomes central in calculus

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Notation and Special Sequences

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

Sigma Notation

• $\sum_{i=m}^{n} f(i)$ is compact notation where $i$ is the index variable, $m$ is the starting value, $n$ is the ending value, and $f(i)$ defines each term
• It translates "add up all terms from $i = m$ to $i = n$" into a single expression
• Useful properties: you can factor out constants ($\sum c \cdot f(i) = c \cdot \sum f(i)$), split sums across addition, and shift indices

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.
• The sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34... Memorize the first several terms for quick recognition on exams.
• Fibonacci is a second-order recurrence relation because it depends on two previous terms, not just one. This makes it more complex than a standard recursive sequence but creates richer patterns.

Arithmetic-Geometric Sequence Formula

• $a_n = (a_1 + (n-1)d) \cdot r^{(n-1)}$ combines linear growth with exponential growth in a single term
• The arithmetic component $(a_1 + (n-1)d)$ gets multiplied by the geometric component $r^{(n-1)}$
• This hybrid structure models scenarios like increasing payments with interest, which comes up 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 for each?

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 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?