Sequences & Series Cheat Sheet

A printable reference covering arithmetic sequences, geometric sequences, recursive rules, sigma notation, finite series, and infinite geometric series for grades 9-11.

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Sequences and series help students describe patterns, make predictions, and add repeated terms efficiently. This cheat sheet covers arithmetic and geometric sequences, recursive definitions, explicit formulas, and sigma notation. Students in grades 9 through 11 need these tools for algebra, functions, finance, and later precalculus topics. A clear formula reference makes it easier to identify the pattern before choosing a method. The main idea is to decide whether terms change by a common difference $d$ or a common ratio $r$ . Arithmetic sequences use $a_n = a_1 + (n - 1)d$ , while geometric sequences use $a_n = a_1r^{n - 1}$ . Series formulas find sums, such as $S_n = \frac{n}{2}(a_1 + a_n)$ for arithmetic series and $S_n = a_1\frac{1 - r^n}{1 - r}$ for geometric series. Infinite geometric series only have a finite sum when $|r| < 1$ .

Key Facts

An arithmetic sequence has a constant difference $d$ , so $a_n = a_1 + (n - 1)d$ .
The common difference in an arithmetic sequence is $d = a_n - a_{n - 1}$ .
A geometric sequence has a constant ratio $r$ , so $a_n = a_1r^{n - 1}$ .
The common ratio in a geometric sequence is $r = \frac{a_n}{a_{n - 1}}$ when $a_{n - 1} \ne 0$ .
The sum of the first $n$ terms of an arithmetic series is $S_n = \frac{n}{2}(a_1 + a_n)$ .
The sum of the first $n$ terms of a geometric series is $S_n = a_1\frac{1 - r^n}{1 - r}$ when $r \ne 1$ .
An infinite geometric series has sum $S_\infty = \frac{a_1}{1 - r}$ only when $|r| < 1$ .
Sigma notation $\sum_{k = m}^{n} a_k$ means add the terms $a_m + a_{m + 1} + \cdots + a_n$ .

Vocabulary

Sequence: A sequence is an ordered list of numbers, often written as terms such as $a_1, a_2, a_3, \ldots$ .
Term: A term is one number in a sequence, and $a_n$ represents the term in position $n$ .
Arithmetic Sequence: An arithmetic sequence is a sequence where each term is found by adding the same difference $d$ to the previous term.
Geometric Sequence: A geometric sequence is a sequence where each term is found by multiplying the previous term by the same ratio $r$ .
Series: A series is the sum of the terms of a sequence, often written using $S_n$ or sigma notation.
Sigma Notation: Sigma notation uses $\sum$ to represent repeated addition of terms from a starting index to an ending index.

Common Mistakes to Avoid

Confusing $d$ and $r$ is wrong because arithmetic patterns add the same amount while geometric patterns multiply by the same factor.
Using $a_n = a_1 + nd$ instead of $a_n = a_1 + (n - 1)d$ is wrong because the first term already occurs when $n = 1$ .
Forgetting the condition $|r| < 1$ for $S_\infty = \frac{a_1}{1 - r}$ is wrong because infinite geometric series diverge when the terms do not shrink toward $0$ .
Starting sigma notation at the wrong index is wrong because $\sum_{k = 1}^{n} a_k$ and $\sum_{k = 0}^{n} a_k$ include different numbers of terms.
Using the geometric series formula when $r = 1$ is wrong because $S_n = a_1\frac{1 - r^n}{1 - r}$ would require division by $0$ .

Practice Questions

1 Find $a_{12}$ for the arithmetic sequence with $a_1 = 7$ and $d = 4$ .
2 Find $S_8$ for the geometric series with $a_1 = 3$ and $r = 2$ .
3 Evaluate $\sum_{k = 1}^{5} (2k + 1)$ .
4 A sequence begins $5, 10, 20, 40, \ldots$ . Explain whether it is arithmetic, geometric, or neither, and identify the pattern.