Summation and Sequence Notation

13 min read

Quick Reference

Every definition and theorem traces to Rosen 8e, Section 2.4.

Try This First

Before any notation, work this by hand.

A sequence starts at 3 and each later term is double the one before it: 3, 6, 12, 24, and so on.

1. Write the next two terms.
2. Add up the first five terms.
3. Suppose you wanted the sum of the first one hundred terms. Writing out one hundred numbers and adding them would take a long time. What single piece of information about the pattern would let you skip the writing?

Sequences and Summations

A sequence is an ordered list of numbers produced by a rule. The position in the list matters: the first term, the second term, the third term, and so on. Think of a sequence as a process that takes a position (an integer) and gives back the term that sits at that position.

The list 3, 6, 12, 24, 48 is a sequence. Its rule is “start at 3 and multiply by 2 to step forward.” Because each step multiplies by the same fixed number, this particular kind of sequence has a name: a geometric progression.

A summation answers a different question. Once a sequence is in hand, you often want to add a stretch of its terms. Writing $a_3 + a_4 + a_5 + a_6$ is fine for four terms. For a hundred terms, that becomes unreadable. Summation notation packs the same instruction into one symbol with a starting index, an ending index, and a rule for the term.

These two ideas, sequence and summation, run through the rest of discrete mathematics. Counting arguments, recurrence relations, algorithm running times, and induction proofs all rest on reading and writing them fluently.

Prerequisite Hub

This is the first node in the MATH301 sequence strand, so nothing in the course comes before it. The only thing assumed is comfort with function notation $f(n)$ and with integer arithmetic, both carried in from earlier courses.

Builds on

• (none within MATH301) This node has no in-course prerequisites. It gives the vocabulary that later sequence and summation topics depend on.

Unlocks

• dm-sequences-and-summations The full development of sequences and summations, including recurrence relations and closed forms.
• dm-mathematical-induction Proof by induction, where summation formulas become the standard targets to prove.

Prerequisite Map

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Official Definitions

Each definition below is reproduced faithfully from Rosen 8e, Section 2.4, with the page citation. Read the definition box, then read the example that follows it.

Sequence

A sequence is a function from a subset of the set of integers (usually either the set $\{0, 1, 2, \ldots\}$ or the set $\{1, 2, 3, \ldots\}$) to a set $S$. We use the notation $a_n$ to denote the image of the integer $n$. We call $a_n$ a term of the sequence.

(Rosen 8e, Section 2.4, p.165, verbatim definition box)

The phrase “a function from a subset of the integers” is the heart of it. A sequence is a function whose inputs are positions. The notation $a_n$ is the same idea as $f(n)$, written in the index style that sequences use.

Reading the notation. In $a_n$, the letter $a$ names the sequence and the subscript $n$ is the position. So $a_3$ means “the term at position 3.” The whole sequence can be written $\{a_n\}$.

Geometric progression

A geometric progression is a sequence of the form $a, ar, ar^2, \ldots, ar^n, \ldots$ where the initial term $a$ and the common ratio $r$ are real numbers.

(Rosen 8e, Section 2.4, p.165, verbatim definition box)

The initial term $a$ is where the list starts. The common ratio $r$ is the fixed number every step multiplies by. The term at position $n$ (counting from $n = 0$) is $a_n = a r^n$. The opener’s list 3, 6, 12, 24 is a geometric progression with $a = 3$ and $r = 2$.

Summation notation

The sum $a_m + a_{m+1} + \cdots + a_n$ is written using summation notation as the sum from $j = m$ to $n$ of $a_j$, where $j$ is the index of summation, $m$ is the lower limit, and $n$ is the upper limit; the index of summation runs through all integers from its lower limit $m$ to its upper limit $n$.

(Rosen 8e, Section 2.4, p.168, Summations subsection)

In symbols:

\[ \sum_{j=m}^{n} a_j = a_m + a_{m+1} + \cdots + a_n \]

The three parts to identify every time:

• $j$ is the index of summation, a placeholder that walks through the integers one at a time.
• $m$ is the lower limit, where the index starts.
• $n$ is the upper limit, where the index stops (and $n$ is included).

The index name is arbitrary. $\sum_{j=1}^{4} j^2$ and $\sum_{k=1}^{4} k^2$ mean the exact same number, because the index is only a placeholder.

Two Representations of the Same Sum

A summation is easiest to trust when you can see it expand. Read this small example three ways and translate between them.

Symbolic form:

\[ \sum_{j=1}^{4} (2j + 1) \]

Expanded form (one term per value of the index):

\[ \bigl(2 \cdot 1 + 1\bigr) + \bigl(2 \cdot 2 + 1\bigr) + \bigl(2 \cdot 3 + 1\bigr) + \bigl(2 \cdot 4 + 1\bigr) \]

Table form:

Adding the right column gives $3 + 5 + 7 + 9 = 24$. The symbolic form, the expanded form, and the table are three views of one quantity. Moving from any one of them to the others is the working skill to practice.

Theorem: Sum of a Finite Geometric Progression

This is the formula that answers the opener’s third question: add many terms of a geometric progression without writing them all out.

Theorem 1 (Sum of a finite geometric progression). If $a$ and $r$ are real numbers and $r$ is not equal to $0$, then \[ > \sum_{j=0}^{n} a r^{j} = \begin{cases} \dfrac{a r^{n+1} - a}{r - 1} & \text{when } r \neq 1, \\[2mm] (n + 1)\, a & \text{when } r = 1. \end{cases} > \]

(Rosen 8e, Section 2.4, Theorem 1, p.169)

Reading the two cases. When the ratio $r$ is anything other than $1$, the closed form $\frac{a r^{n+1} - a}{r - 1}$ collapses the whole sum into one fraction. When $r = 1$, every term equals $a$, so the sum is just $a$ added $n + 1$ times, which is $(n + 1)a$. The split is necessary because the denominator $r - 1$ would be zero in the first formula when $r = 1$.

Counting the terms. The index runs $j = 0, 1, 2, \ldots, n$, so there are $n + 1$ terms, not $n$. Off-by-one slips here are the most common source of a wrong answer.

Worked Examples

Example 1: Expanding and evaluating a summation

Evaluate $\displaystyle \sum_{k=2}^{5} k^2$.

Predict first. The terms are squares of 2, 3, 4, 5. The largest term, $25$, dominates, so the total should land somewhere above $25$ and below $4 \times 25 = 100$. A reasonable guess is around $50$ to $60$.

Now compute. The lower limit is $2$ and the upper limit is $5$, so the index $k$ takes the values $2, 3, 4, 5$:

\[ \sum_{k=2}^{5} k^2 = 2^2 + 3^2 + 4^2 + 5^2 = 4 + 9 + 16 + 25 = 54. \]

Compare. The result $54$ sits inside the predicted range. The prediction caught no arithmetic error this time, but doing it builds the habit that catches one later.

Example 2: Identifying the parts of a geometric progression

Consider the sequence $5, 15, 45, 135, \ldots$.

Step 1: Find the common ratio. Divide any term by the one before it: $\frac{15}{5} = 3$ and $\frac{45}{15} = 3$. The common ratio is $r = 3$.

Step 2: Find the initial term. The list starts at $5$, so $a = 5$.

Step 3: Write the general term. Counting positions from $n = 0$, the term at position $n$ is

\[ a_n = a r^n = 5 \cdot 3^{n}. \]

Check. Position $n = 0$ gives $5 \cdot 3^0 = 5$. Position $n = 2$ gives $5 \cdot 3^2 = 45$. Both match the listed terms, so the formula is correct.

Example 3: Summing a geometric progression with the theorem

Find $\displaystyle \sum_{j=0}^{4} 5 \cdot 3^{j}$.

Predict first. This is the sum of $5, 15, 45, 135, 405$. The last term alone is $405$, so the total must exceed $405$. A rough mental estimate is “a bit over $600$.”

Now apply Theorem 1. Here $a = 5$, $r = 3$ (which is not $1$), and the upper limit is $n = 4$:

\[ \sum_{j=0}^{4} 5 \cdot 3^{j} = \frac{a r^{n+1} - a}{r - 1} = \frac{5 \cdot 3^{5} - 5}{3 - 1} = \frac{5 \cdot 243 - 5}{2} = \frac{1215 - 5}{2} = \frac{1210}{2} = 605. \]

Verify against the long way. Adding by hand: $5 + 15 + 45 + 135 + 405 = 605$. The closed form and the direct sum agree, and $605$ matches the prediction.

One way versus another. One way to see this sum is to apply the theorem directly, as above. Another way is to factor the initial term out first: $\sum_{j=0}^{4} 5 \cdot 3^{j} = 5 \sum_{j=0}^{4} 3^{j} = 5 \cdot \frac{3^5 - 1}{3 - 1} = 5 \cdot \frac{242}{2} = 5 \cdot 121 = 605$. The two paths give the same number. Which one feels clearer to you, and why?

Example 4: The $r = 1$ case

Find $\displaystyle \sum_{j=0}^{6} 7 \cdot 1^{j}$.

Here $a = 7$ and $r = 1$. Every term is $7 \cdot 1^j = 7$, and the index runs from $0$ to $6$, which is $7$ terms. The theorem’s second case gives

\[ \sum_{j=0}^{6} 7 \cdot 1^{j} = (n + 1)\,a = (6 + 1)\cdot 7 = 49. \]

Checking directly, $7$ added seven times is $49$. The first-case formula cannot be used here because its denominator $r - 1 = 0$.

Common Misconceptions

Practice Problems

Mastery Checklist

Novice (Level 1-2):

• I can read $a_n$ as the term at position $n$ and evaluate it from a rule.
• I can expand a summation $\sum_{j=m}^{n} a_j$ into its individual terms.
• I can name the index of summation, the lower limit, and the upper limit.

Competent (Level 3-4):

• I can find the initial term and common ratio of a geometric progression and write its general term.
• I can apply the finite geometric sum theorem and check the result by direct addition.
• I can count the number of terms in a summation correctly, including both endpoints.

Proficient (Level 5):

• I can reindex a summation so its lower limit matches the form a theorem requires.
• I can explain why the geometric sum theorem needs a separate case for $r = 1$.

Mental Model

A sequence is a vending machine for numbers: you press a position (an integer), and it dispenses the term at that position. The rule inside the machine is the same whether you press position $2$ or position $200$.

A summation is a request to the machine for a range of positions at once, followed by adding everything that comes out. Summation notation is the order form: lower limit, upper limit, and the rule for each term.

The geometric sum theorem is a shortcut printed on the side of the machine for one special kind of rule (fixed start, fixed multiplier), so you never have to collect and add a long stretch of terms by hand.

Resources

• Primary text: Rosen, Discrete Mathematics and Its Applications, 8th ed., Section 2.4 Sequences and Summations, pages 165 to 171. The verbatim definitions and Theorem 1 used on this page come from here.
• Open companion: Levin, Discrete Mathematics: An Open Introduction (3rd ed.), the Sequences chapter, a freely available treatment of the same ideas. discrete.openmathbooks.org/dmoi3/ch_sequences.html
• In-tree continuation: Sequences and Summations (dm-sequences-and-summations), the development that builds recurrences and closed forms on this vocabulary.

Connections

Looking back:

• This is the first node in the MATH301 sequence strand, so nothing in the course comes before it. It assumes only function notation $f(n)$ and integer arithmetic from earlier courses.

Looking ahead:

• Sequences and Summations: the full treatment, including recurrence relations and finding closed forms.
• Mathematical Induction (dm-mathematical-induction): summation identities become the standard statements proved by induction, where the formula on the right is the target.

Real-world connections:

• Compound interest grows as a geometric progression: a fixed principal multiplied by a fixed growth factor each period.
• The running time of an algorithm that halves its input at each step is read off a geometric sum.
• Population models with a constant per-period growth rate produce geometric sequences of counts.

Last updated: 2026-06-16