Series Formulas

Why This Matters

Series formulas are the backbone of pattern recognition in mathematics—they transform scattered sequences into predictable, calculable structures. In Honors Pre-Calc, you're being tested on your ability to recognize which type of series you're dealing with, which formula applies, and how to manipulate these formulas to find terms, sums, and limits. These concepts connect directly to calculus (where series become infinite and convergence matters) and to real-world modeling in finance, physics, and computer science.

Don't just memorize formulas—understand what each formula does and when to use it. Can you identify an arithmetic vs. geometric pattern on sight? Do you know why an infinite geometric series only converges under certain conditions? These are the questions that separate strong test performance from struggling through FRQs. Master the underlying logic, and the formulas become tools rather than obstacles.

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Arithmetic Patterns: Constant Addition

Arithmetic sequences and series are built on constant differences—each term grows (or shrinks) by the same amount. This linear growth pattern is the simplest to identify and the most common in word problems involving regular payments, distances, or schedules.

Arithmetic Sequence Formula

• The nth term formula $a_n = a_1 + (n-1)d$—where $a_1$ is the first term and $d$ is the common difference
• Common difference (d) is found by subtracting any term from the next: $d = a_{n+1} - a_n$
• Linear relationship means graphing terms against their position produces a straight line—useful for checking your work

Arithmetic Series Sum Formula

• Sum of n terms uses $S_n = \frac{n}{2}(a_1 + a_n)$ or equivalently $S_n = \frac{n}{2}(2a_1 + (n-1)d)$
• The "pairing" logic comes from Gauss's insight: first + last = second + second-to-last, giving $\frac{n}{2}$ pairs
• Applications include total salary over time, accumulated distance, or summing consecutive integers

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Geometric Patterns: Constant Multiplication

Geometric sequences and series involve constant ratios—each term is multiplied by the same factor. This exponential behavior models growth and decay in finance, populations, and radioactive materials.

Geometric Sequence Formula

• The nth term formula $a_n = a_1 \cdot r^{(n-1)}$—where $r$ is the common ratio between consecutive terms
• Common ratio (r) is found by dividing any term by the previous: $r = \frac{a_{n+1}}{a_n}$
• Exponential growth occurs when $|r| > 1$; exponential decay occurs when $0 < |r| < 1$

Geometric Series Sum Formula

• Finite sum formula is $S_n = a_1 \cdot \frac{1 - r^n}{1 - r}$ for $r \neq 1$
• Watch the constraint—when $r = 1$, every term equals $a_1$, so $S_n = n \cdot a_1$ instead
• Investment applications use this for compound interest calculations and annuity totals

Infinite Geometric Series Sum Formula

• Convergence requirement is $|r| < 1$—only then does the infinite sum exist
• The formula simplifies to $S = \frac{a_1}{1 - r}$ when the series converges
• Divergence occurs when $|r| \geq 1$, meaning the sum grows without bound or oscillates

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Special Sequence Types

Some sequences don't fit neatly into arithmetic or geometric categories. These hybrid and specialized patterns appear in advanced problems and require recognizing their unique structures.

Harmonic Sequence Formula

• Defined as reciprocals of an arithmetic sequence: $a_n = \frac{1}{a_1 + (n-1)d}$
• The harmonic series $\sum \frac{1}{n}$ famously diverges—it grows infinitely, just very slowly
• Physics applications include wave frequencies, resistors in parallel, and lens equations

Arithmetic-Geometric Sequence Formula

• Hybrid structure multiplies an arithmetic term by a geometric term: $a_n = (a + (n-1)d) \cdot r^{n-1}$
• Summing requires special techniques—multiply the series by $r$ and subtract to eliminate terms
• Appears in problems combining linear growth with exponential factors, like increasing payments with interest

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Telescoping and Simplification Techniques

Telescoping series exploit cancellation patterns to make complex sums simple. Recognizing when terms cancel is a powerful problem-solving strategy.

Telescoping Series Formula

• Cancellation structure means most terms vanish, leaving only $S_n = a_1 - a_{n+1}$ (or similar boundary terms)
• Partial fractions often reveal telescoping: $\frac{1}{n(n+1)} = \frac{1}{n} - \frac{1}{n+1}$
• Limit evaluation becomes straightforward once you identify the surviving terms

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Binomial Expansion Tools

The binomial theorem and Pascal's Triangle provide systematic expansion methods for expressions like $(a + b)^n$. These connect series to combinatorics and probability.

Binomial Theorem

• Expansion formula is $(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$
• Binomial coefficients $\binom{n}{k} = \frac{n!}{k!(n-k)!}$ count combinations and determine term weights
• Probability connections—binomial distributions use these coefficients to calculate outcome likelihoods

Pascal's Triangle

• Construction rule—each entry equals the sum of the two entries directly above it
• Row n contains the coefficients for $(a + b)^n$: row 4 gives $1, 4, 6, 4, 1$ for $(a+b)^4$
• Symmetry property means $\binom{n}{k} = \binom{n}{n-k}$—useful for simplifying calculations

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Quick Reference Table

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Self-Check Questions

1. What is the key difference between when you use $S_n = a_1 \cdot \frac{1 - r^n}{1 - r}$ versus $S = \frac{a_1}{1 - r}$, and what condition must you verify?

2. Given the sequence $3, 6, 12, 24, ...$, identify whether it's arithmetic or geometric, state the common difference or ratio, and write the formula for the nth term.

3. Compare and contrast how you would find the sum of the first 50 terms of an arithmetic series versus the sum of an infinite geometric series with $r = 0.5$.

4. If a telescoping series has the form $\sum_{n=1}^{N} \left(\frac{1}{n} - \frac{1}{n+1}\right)$, what is the simplified sum, and which terms survive?

5. Using either the Binomial Theorem or Pascal's Triangle, what is the coefficient of $a^2b^3$ in the expansion of $(a + b)^5$?