Sequences & Series Cheat Sheet

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.

Understanding Sequences & Series

A sequence is an ordered list, so position is part of the meaning. The fifth term is not simply a value from the list. It is the value attached to position five.

This matters when a rule begins at zero instead of one. In computer science and some graphing contexts, counting often starts at zero. Students should always check the first index before substituting a term number into a rule.

A recursive rule gives a starting value and a process for producing the next value. It is useful for building a table step by step.

An explicit rule gives any requested term directly. It is usually faster when the problem asks for a distant term, such as the one hundredth term.

Arithmetic patterns model steady change. A savings account receiving the same dollar deposit each week can form an arithmetic sequence if interest is ignored. The height of stacked equal blocks and the cost of a taxi with a fixed charge per mile can have the same kind of linear growth.

The graph of term number against term value forms points on a straight line. A frequent mistake is using the term number itself instead of one less than the term number. The first term needs zero changes from the starting value.

Another common error is confusing a negative difference with a sequence that has negative terms. A sequence can begin positive and decrease by the same amount until its terms become negative.

Geometric patterns describe repeated multiplication, which creates exponential change. Population models, compound interest, depreciation, image resizing, and repeated discounts can involve geometric sequences. The ratio can be a fraction, which causes decay, or a negative number, which makes signs alternate.

For example, multiplying repeatedly by negative one-half produces terms that switch from positive to negative while getting closer to zero. Zero needs care in geometric work.

Once a term becomes zero, later terms can remain zero, but dividing by a previous zero term cannot determine a ratio. Students should compare several pairs of neighboring terms before deciding that a pattern is geometric, especially when the list includes fractions or negative values.

A series adds terms, and the number of included terms must be clear. Sigma notation is compact, but it can hide counting errors. Write out the first few and last few terms before evaluating a sum.

If the index starts at three and ends at ten, there are eight terms, not seven. For arithmetic sums, pairing the first and last terms explains why the average term is useful. Each pair has the same total, even when the number of terms is odd and one middle term is left.

Infinite geometric sums are about a limit, not about literally finishing infinitely many additions. When repeated multiplication makes the terms approach zero, the partial sums can settle near one fixed value. If the terms fail to shrink, a finite infinite sum does not exist.

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