Sequences and Series

You already spot patterns all the time without thinking about it. If someone gave you the list 2, 4, 6, 8 and asked what comes next, you would say 10 without hesitation. You recognize that each number is 2 more than the one before it. That intuition - the ability to see structure in a list of numbers - is exactly what sequences and series are about.

A sequence is simply an ordered list of numbers that follows some rule. The numbers 2, 4, 6, 8, … form a sequence where each term is 2 more than the previous one. The numbers 3, 6, 12, 24, … form a sequence where each term is double the previous one. Once you identify the pattern, you can predict any term in the list, even the 100th or 1000th term, without writing out every number in between.

A series takes this one step further: instead of just listing the terms, you add them up. The sequence 2, 4, 6, 8 becomes the series 2 + 4 + 6 + 8. This distinction matters because many real-world problems ask not just “what is the pattern?” but “what is the total?”

Think about saving money. If you save $100 in January, $150 in February, $200 in March, and so on (adding $50 each month), the amounts form a sequence. But the question you probably care about is: how much will you have saved after a year? That requires adding up all the terms - that is a series.

Core Concepts

Sequences: The Foundation

A sequence is an ordered list of numbers, called terms, that follow a specific pattern or rule. We use subscript notation to identify individual terms:

• $a_1$ is the first term
• $a_2$ is the second term
• $a_n$ is the nth term (the general term)

For example, in the sequence 5, 8, 11, 14, …:

• $a_1 = 5$
• $a_2 = 8$
• $a_3 = 11$
• $a_4 = 14$

The “…” (ellipsis) indicates that the sequence continues following the same pattern.

Arithmetic Sequences: Adding the Same Amount

An arithmetic sequence is one where you get from one term to the next by adding (or subtracting) the same number each time. This constant amount is called the common difference, denoted by $d$.

In the sequence 7, 11, 15, 19, 23, …:

• From 7 to 11, you add 4
• From 11 to 15, you add 4
• From 15 to 19, you add 4

The common difference is $d = 4$.

To find the common difference, subtract any term from the term that follows it:

$$d = a_{n+1} - a_n$$

If the common difference is positive, the sequence increases. If it is negative, the sequence decreases. The sequence 20, 17, 14, 11, … has a common difference of $d = -3$.

Geometric Sequences: Multiplying by the Same Amount

A geometric sequence is one where you get from one term to the next by multiplying by the same number each time. This constant multiplier is called the common ratio, denoted by $r$.

In the sequence 3, 6, 12, 24, 48, …:

• From 3 to 6, you multiply by 2
• From 6 to 12, you multiply by 2
• From 12 to 24, you multiply by 2

The common ratio is $r = 2$.

To find the common ratio, divide any term by the term before it:

$$r = \frac{a_{n+1}}{a_n}$$

If $r > 1$, the terms grow larger. If $0 < r < 1$, the terms shrink toward zero. If $r < 0$, the terms alternate between positive and negative. The sequence 81, 27, 9, 3, 1, … has a common ratio of $r = \frac{1}{3}$.

Explicit vs. Recursive Formulas

There are two ways to describe a sequence with a formula:

Explicit formulas give you the nth term directly. You plug in $n$ and get the answer without needing to know any other terms. For an arithmetic sequence:

$$a_n = a_1 + (n - 1)d$$

For example, in the sequence 5, 8, 11, 14, … where $a_1 = 5$ and $d = 3$:

$$a_n = 5 + (n - 1)(3) = 5 + 3n - 3 = 3n + 2$$

To find $a_{50}$, just substitute: $a_{50} = 3(50) + 2 = 152$. No need to list all 50 terms.

Recursive formulas tell you how to get each term from the previous one. They require knowing a starting value plus a rule for generating successive terms.

For an arithmetic sequence: $$a_1 = \text{(first term)}$$ $$a_n = a_{n-1} + d$$

For the sequence 5, 8, 11, 14, … this would be: $$a_1 = 5$$ $$a_n = a_{n-1} + 3$$

Recursive formulas are intuitive (they describe how the sequence actually builds) but impractical for finding distant terms. Explicit formulas let you jump directly to any term.

The nth Term of an Arithmetic Sequence

The explicit formula for the nth term of an arithmetic sequence is:

$$a_n = a_1 + (n - 1)d$$

This formula makes sense if you think about it: to get from $a_1$ to $a_n$, you add the common difference $d$ exactly $(n - 1)$ times. To get to the 10th term, you add $d$ nine times. To get to the 100th term, you add $d$ ninety-nine times.

The nth Term of a Geometric Sequence

The explicit formula for the nth term of a geometric sequence is:

$$a_n = a_1 \cdot r^{n-1}$$

Again, this follows from how the sequence builds: to get from $a_1$ to $a_n$, you multiply by $r$ exactly $(n - 1)$ times. If $a_1 = 2$ and $r = 3$:

• $a_1 = 2$
• $a_2 = 2 \cdot 3 = 6$
• $a_3 = 2 \cdot 3^2 = 18$
• $a_n = 2 \cdot 3^{n-1}$

Series: Adding It Up

A series is the sum of the terms of a sequence. If your sequence is $a_1, a_2, a_3, …, a_n$, then the corresponding series is:

$$a_1 + a_2 + a_3 + … + a_n$$

We often use $S_n$ to denote the sum of the first $n$ terms.

Arithmetic Series: The Sum Formula

Adding up an arithmetic sequence by hand is tedious for large $n$. Fortunately, there is a shortcut. The sum of the first $n$ terms of an arithmetic sequence is:

$$S_n = \frac{n}{2}(a_1 + a_n)$$

Or equivalently:

$$S_n = \frac{n}{2}(2a_1 + (n-1)d)$$

The first formula is easier to remember if you know both the first and last terms. The second is useful when you only know $a_1$, $d$, and $n$.

Here is the intuition: if you pair the first term with the last, the second term with the second-to-last, and so on, each pair has the same sum. For the sequence 1, 2, 3, 4, 5, 6, 7, 8, 9, 10:

• $1 + 10 = 11$
• $2 + 9 = 11$
• $3 + 8 = 11$
• $4 + 7 = 11$
• $5 + 6 = 11$

There are 5 pairs (which is $\frac{10}{2}$), each summing to 11. Total: $5 \times 11 = 55$.

Geometric Series: The Sum Formula

The sum of the first $n$ terms of a geometric sequence (where $r \neq 1$) is:

$$S_n = a_1 \cdot \frac{1 - r^n}{1 - r}$$

Or equivalently:

$$S_n = a_1 \cdot \frac{r^n - 1}{r - 1}$$

Both forms give the same result; use whichever feels more natural for the values you are working with.

Infinite Geometric Series

Something remarkable happens with geometric sequences when the common ratio $r$ is between $-1$ and $1$ (that is, when $|r| < 1$). As you go further into the sequence, the terms get smaller and smaller, approaching zero. This means you can actually add up infinitely many terms and get a finite answer.

Consider the series $1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + …$

Each term is half the previous one. If you keep adding forever, the sum approaches 2 (not infinity). This is an infinite geometric series.

When $|r| < 1$, the sum of an infinite geometric series is:

$$S_\infty = \frac{a_1}{1 - r}$$

For the series above, $a_1 = 1$ and $r = \frac{1}{2}$:

$$S_\infty = \frac{1}{1 - \frac{1}{2}} = \frac{1}{\frac{1}{2}} = 2$$

If $|r| \geq 1$, the terms do not shrink toward zero, and the infinite sum does not exist (or is infinite).

Sigma Notation: Compact Summation

Writing out a long sum like $1 + 2 + 3 + … + 100$ is tedious. Mathematicians use sigma notation (named after the Greek letter $\Sigma$) as a compact way to write sums.

$$\sum_{k=1}^{n} a_k = a_1 + a_2 + a_3 + … + a_n$$

The parts of sigma notation:

• $\Sigma$ means “sum”
• $k = 1$ below tells you the starting index
• $n$ above tells you the ending index
• $a_k$ is the expression you are summing

For example: $$\sum_{k=1}^{5} k = 1 + 2 + 3 + 4 + 5 = 15$$

$$\sum_{i=1}^{4} 2^i = 2^1 + 2^2 + 2^3 + 2^4 = 2 + 4 + 8 + 16 = 30$$

The index variable ($k$, $i$, $n$, or any letter) is just a placeholder - it does not matter what you call it.

Notation and Terminology

Term	Meaning	Example
Sequence	Ordered list of numbers following a pattern	2, 5, 8, 11, …
Series	Sum of the terms in a sequence	2 + 5 + 8 + 11 + …
$a_n$	The nth term of a sequence	$a_3 = 8$ (third term is 8)
Common difference $d$	Constant added between consecutive terms (arithmetic)	$d = 3$ in 2, 5, 8, 11, …
Common ratio $r$	Constant multiplied between consecutive terms (geometric)	$r = 2$ in 3, 6, 12, 24, …
$S_n$	Sum of the first $n$ terms	$S_4 = 2 + 5 + 8 + 11 = 26$
$\sum$ (sigma)	Summation notation	$\sum_{i=1}^{5} i = 1+2+3+4+5$
Explicit formula	Formula that gives $a_n$ directly	$a_n = 3n + 2$
Recursive formula	Formula relating $a_n$ to previous terms	$a_n = a_{n-1} + 3$

Examples

Key Properties and Rules

Formulas Summary

Arithmetic Sequences and Series:

Formula	Purpose
$a_n = a_1 + (n-1)d$	nth term of arithmetic sequence
$d = a_{n+1} - a_n$	Common difference
$S_n = \frac{n}{2}(a_1 + a_n)$	Sum of first n terms (when you know $a_n$)
$S_n = \frac{n}{2}(2a_1 + (n-1)d)$	Sum of first n terms (when you know $d$)

Geometric Sequences and Series:

Formula	Purpose
$a_n = a_1 \cdot r^{n-1}$	nth term of geometric sequence
$r = \frac{a_{n+1}}{a_n}$	Common ratio
$S_n = a_1 \cdot \frac{1-r^n}{1-r}$	Sum of first n terms ($r \neq 1$)
$S_\infty = \frac{a_1}{1-r}$	Infinite sum (only when $|r| < 1$)

Identifying Sequence Types

To determine if a sequence is arithmetic:

• Calculate the differences between consecutive terms
• If all differences are equal, it is arithmetic

To determine if a sequence is geometric:

• Calculate the ratios between consecutive terms
• If all ratios are equal, it is geometric

Some sequences are neither arithmetic nor geometric. The Fibonacci sequence (1, 1, 2, 3, 5, 8, …) is a famous example - each term is the sum of the two preceding terms, but there is no common difference or ratio.

Sigma Notation Properties

These properties let you manipulate sums algebraically:

$$\sum_{k=1}^{n} c \cdot a_k = c \cdot \sum_{k=1}^{n} a_k$$ (constants can be factored out)

$$\sum_{k=1}^{n} (a_k + b_k) = \sum_{k=1}^{n} a_k + \sum_{k=1}^{n} b_k$$ (sums can be split)

$$\sum_{k=1}^{n} c = nc$$ (summing a constant n times gives nc)

Common Mistakes to Avoid

• Using $n$ instead of $(n-1)$ in the explicit formulas. To get to the nth term, you add $d$ (or multiply by $r$) only $(n-1)$ times, not $n$ times.
• Forgetting to check $|r| < 1$ before using the infinite series formula. If $|r| \geq 1$, the infinite sum does not exist.
• Confusing sequences and series. A sequence is a list; a series is a sum. The formula $a_n = a_1 + (n-1)d$ gives you a single term. The formula $S_n = \frac{n}{2}(a_1 + a_n)$ gives you a sum.
• Misidentifying the sequence type. Always verify the pattern by checking multiple terms. A sequence might look arithmetic at first glance but turn out to be something else.

Real-World Applications

Saving Plans (Arithmetic)

If you deposit $500 initially and add $100 each month, your deposits form an arithmetic sequence:

$$500, 600, 700, 800, …$$

After 12 months, your total deposits are: $$S_{12} = \frac{12}{2}(500 + 1600) = 6 \times 2100 = \$12{,}600$$

The arithmetic series formula saves you from adding twelve numbers by hand.

Compound Interest (Geometric)

If you invest $1000 at 5% annual interest compounded yearly, your balance each year forms a geometric sequence:

$$1000, 1050, 1102.50, 1157.63, …$$

The common ratio is $r = 1.05$ (representing 100% of the previous balance plus 5% interest). After 20 years: $$a_{20} = 1000 \cdot (1.05)^{19} \approx \$2{,}526.95$$

The geometric sequence formula explains why compound interest grows so dramatically over time.

Bouncing Ball Distance

A ball dropped from 10 meters bounces back to 60% of its previous height each time. The heights form a geometric sequence:

$$10, 6, 3.6, 2.16, …$$

The total vertical distance traveled (down and up) converges because $r = 0.6 < 1$. Using the infinite series formula for just the downward distances: $$S_\infty = \frac{10}{1 - 0.6} = 25 \text{ meters}$$

Including both directions, the ball travels $2(25) - 10 = 40$ meters total (subtracting the initial drop counted twice).

Annuities and Loans

Regular payments into a retirement account or regular payments on a mortgage involve geometric series. The future value of regular deposits and the amortization of loans both rely on the geometric series sum formula - which is why financial calculators and spreadsheets have these formulas built in.

Fractal Patterns

Fractals like the Koch snowflake or Sierpinski triangle involve infinite geometric series. Each iteration adds smaller and smaller pieces, but because the ratio is less than 1, the total area or perimeter converges to a finite value.

Self-Test Problems

Summary

• A sequence is an ordered list of numbers following a pattern. A series is the sum of those numbers.
• Arithmetic sequences have a constant difference $d$ between consecutive terms. The nth term is $a_n = a_1 + (n-1)d$, and the sum of the first $n$ terms is $S_n = \frac{n}{2}(a_1 + a_n)$.
• Geometric sequences have a constant ratio $r$ between consecutive terms. The nth term is $a_n = a_1 \cdot r^{n-1}$, and the sum of the first $n$ terms is $S_n = a_1 \cdot \frac{1-r^n}{1-r}$.
• Explicit formulas give you any term directly. Recursive formulas build each term from the previous one.
• Infinite geometric series converge to a finite sum only when $|r| < 1$. That sum is $S_\infty = \frac{a_1}{1-r}$.
• Sigma notation ($\Sigma$) provides a compact way to write sums, with the index, bounds, and formula all specified.
• Sequences and series appear throughout mathematics and real life: savings plans, compound interest, loan payments, bouncing balls, fractals, and countless other applications.

The patterns you learned to spot as a child - what comes next in a list - turn out to be surprisingly powerful when you formalize them. With explicit formulas, you can find the millionth term without listing a million numbers. With sum formulas, you can add thousands of terms in one calculation. That is the power of recognizing patterns and expressing them mathematically.