Key Concepts of Sequences and Series

Why This Matters

Sequences and series form the backbone of understanding patterns in mathematics—and they show up everywhere on exams. You're being tested on your ability to recognize how numbers grow (linearly vs. exponentially), predict future terms, and calculate sums efficiently. These concepts connect directly to functions, exponential growth models, and even calculus-readiness topics like limits and convergence.

Don't just memorize formulas—know why each formula works and when to apply it. Can you tell the difference between a recursive and explicit formula at a glance? Do you know which series converge and which blow up to infinity? Master the underlying logic, and you'll handle any problem they throw at you.

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Defining Sequences: How Terms Are Generated

Every sequence has a rule that determines its terms. The key distinction is how that rule is expressed—either by relating each term to the previous one, or by giving a direct formula for any term.

Recursive Formulas

• Defines each term using previous terms—you need the term before to find the next one
• Arithmetic example: $a_n = a_{n-1} + d$ builds each term by adding the common difference
• Best for: generating terms step-by-step or when the pattern is easier to describe relationally

Explicit Formulas

• Calculates any term directly without needing previous terms—plug in $n$ and get your answer
• Arithmetic: $a_n = a_1 + (n-1)d$; Geometric: $a_n = a_1 \cdot r^{(n-1)}$
• Best for: finding the 100th term without calculating the first 99

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Arithmetic Patterns: Constant Addition

Arithmetic sequences grow by adding the same value repeatedly. This creates linear growth—graph the terms and you'll see a straight line.

Arithmetic Sequences

• Common difference ($d$) is added to each term to get the next: $a_n = a_1 + (n-1)d$
• Linear growth pattern—the sequence increases (or decreases) at a constant rate
• Identify by: subtracting consecutive terms; if the difference is always the same, it's arithmetic

Arithmetic Series

• Sum of arithmetic sequence terms using $S_n = \frac{n}{2}(a_1 + a_n)$ or $S_n = \frac{n}{2}(2a_1 + (n-1)d)$
• Two formula versions—use the first when you know the last term, the second when you don't
• Visual interpretation: the sum equals the average of first and last terms, multiplied by the number of terms

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Geometric Patterns: Constant Multiplication

Geometric sequences grow by multiplying by the same value repeatedly. This creates exponential growth (or decay), which behaves very differently from arithmetic patterns.

Geometric Sequences

• Common ratio ($r$) multiplies each term to get the next: $a_n = a_1 \cdot r^{(n-1)}$
• Exponential behavior—grows rapidly if $|r| > 1$, shrinks toward zero if $|r| < 1$
• Identify by: dividing consecutive terms; if the ratio is constant, it's geometric

Geometric Series

• Sum of geometric sequence terms using $S_n = a_1 \cdot \frac{1 - r^n}{1 - r}$ (when $r \neq 1$)
• The formula fails when $r = 1$—in that case, every term equals $a_1$, so $S_n = n \cdot a_1$
• Watch the signs: when $r$ is negative, terms alternate positive and negative

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Infinite Series: When Sums Have Limits

What happens when a series continues forever? Sometimes the sum approaches a finite value (converges), and sometimes it doesn't (diverges). The common ratio determines everything.

Convergent and Divergent Series

• Convergent series approach a finite sum as $n \to \infty$—the terms shrink fast enough to "settle down"
• Divergent series have no finite sum—they either grow without bound or oscillate indefinitely
• Key test: for geometric series, convergence depends entirely on whether $|r| < 1$

Infinite Geometric Series

• Converges to $S = \frac{a_1}{1 - r}$ when $|r| < 1$—the terms become negligibly small
• Diverges when $|r| \geq 1$—terms don't shrink, so the sum keeps growing or oscillating
• Classic example: $\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots = 1$ (here $a_1 = \frac{1}{2}$, $r = \frac{1}{2}$)

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Notation and Partial Sums: The Language of Series

Efficient notation lets you express complex sums compactly. Mastering sigma notation and understanding partial sums is essential for reading and writing series problems fluently.

Sigma Notation

• Compact representation of sums: $\sum_{i=1}^{n} a_i$ means $a_1 + a_2 + \cdots + a_n$
• Index variable ($i$) counts through terms; the bottom shows where to start, the top shows where to stop
• Translating is key: practice converting between expanded form and sigma notation in both directions

Partial Sums

• The sum of the first $n$ terms, denoted $S_n$—a "snapshot" of the series at term $n$
• Tracks series behavior: watching how $S_n$ changes as $n$ increases reveals convergence or divergence
• Foundation for limits: if $S_n$ approaches a fixed value as $n \to \infty$, the series converges to that value

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Quick Reference Table

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Self-Check Questions

1. What is the key difference between arithmetic and geometric sequences, and how can you identify each from a list of terms?

2. Given the sequence 3, 6, 12, 24, ..., write both the recursive formula and the explicit formula. Which would you use to find the 20th term?

3. Compare the formulas for finite geometric series and infinite geometric series. Under what condition can you use the infinite series formula?

4. If a geometric series has $a_1 = 10$ and $r = -0.5$, does the series converge or diverge? If it converges, what is the sum?

5. Translate $\sum_{k=1}^{5} (2k + 1)$ into expanded form and calculate the sum. Then verify using the arithmetic series formula.