A sequence is an ordered list of numbers with a rule; a series is what you get when you add the list up. From this simple pair of ideas comes recursion, infinite sums with finite values, and the first real taste of what "convergence" means.
13 min readPrereqs: Variables & expressions, Exponents & RadicalsUpdated 2026·05·17
What you'll leave with
The two ways to define a sequence — recursive (next from previous) and closed-form (term directly from $n$) — and how to convert between them.
The two families that cover most of pre-calculus: arithmetic ($a_n = a_1 + (n-1)d$) and geometric ($a_n = a_1 r^{n-1}$).
The arithmetic-series sum via Gauss's pairing trick, and the geometric-series sum via the "shift-and-subtract" trick.
The infinite geometric series $\dfrac{a}{1-r}$, when it converges ($|r| < 1$), and why $0.\overline{9} = 1$ falls out of it.
A first informal picture of convergence, and why the harmonic series breaks the obvious-looking rule.
1. Sequences as functions on the naturals
Sequence
An ordered list of numbers $a_1, a_2, a_3, \ldots$ indexed by the positive integers. Equivalently, a function $a : \mathbb{N} \to \mathbb{R}$ that maps each position $n$ to its term $a_n$.
The word "ordered" carries the whole idea. $\{1, 2, 3\}$ as a set is the same as $\{3, 1, 2\}$, but as sequences they are completely different objects: the first is $a_1=1, a_2=2, a_3=3$; the second is $a_1=3, a_2=1, a_3=2$. The index matters because the index is the input to the function.
A sequence can be finite (ends at some $a_N$) or infinite (continues forever). In this page we mostly mean infinite, but every finite-sum formula we write works just as well for the first $N$ terms of an infinite sequence.
Sequence vs series
A sequence is a list: $1, \tfrac{1}{2}, \tfrac{1}{3}, \ldots$. A series is what you get when you put plus signs between the entries: $1 + \tfrac{1}{2} + \tfrac{1}{3} + \cdots$. They look almost the same on the page and are constantly confused, but they are different objects with different behaviours — the sequence above has limit $0$, while the series (called the harmonic series) actually grows without bound.
2. Two ways to define a sequence
There are two equally legitimate ways to pin down what $a_n$ is.
Recursive definition
Give a starting value (or several) and a rule that produces each new term from the previous one(s).
$$ a_1 = 5, \qquad a_{n+1} = a_n + 4 $$
To find $a_4$ you have to walk: $a_1 = 5$, $a_2 = 9$, $a_3 = 13$, $a_4 = 17$. The rule is easy to state but you can't skip ahead.
Closed-form (explicit) definition
Give a formula for $a_n$ directly in terms of $n$.
$$ a_n = 4n + 1 $$
Now $a_{1000}$ is one line of arithmetic: $4(1000) + 1 = 4001$. The closed-form is harder to discover but much faster to evaluate.
For the two families we're about to meet — arithmetic and geometric — converting between recursive and closed-form is mechanical. For other recurrences (Fibonacci, $F_{n+1} = F_n + F_{n-1}$, comes to mind) it is genuinely hard and a topic for later.
Why bother with both?
Recursive definitions match how processes actually unfold — month-by-month balances, generation-by-generation population counts. Closed-form definitions are how you predict the state at a far-future time without simulating every step in between. Whenever you have a recursive definition and can find its closed-form, you've essentially solved the dynamics.
3. Arithmetic sequences
Arithmetic sequence
A sequence in which the difference between any two consecutive terms is the same constant $d$, called the common difference.
The closed-form is almost an identity once you see it: starting at $a_1$ and adding $d$ a total of $n-1$ times lands you at the $n$-th term. The trickiest thing about it is the off-by-one — the exponent on $d$ is $n-1$, not $n$, because the first term has had zero increments applied to it.
$1, 1, 2, 3, 5, \ldots$ — Fibonacci; neither difference nor ratio constant
To test whether a sequence is arithmetic, compute differences of consecutive terms. If they're all the same number, you have your $d$ and can write the formula immediately.
4. Summing an arithmetic series: Gauss's trick
A famous (probably apocryphal) story: a teacher trying to keep the seven-year-old Carl Friedrich Gauss busy told the class to add the integers from 1 to 100. Gauss produced the answer in a few seconds. The trick he used solves any arithmetic series.
Write the sum forward and then write it again backward:
Now add them column by column. In an arithmetic sequence each column sums to the same thing — $a_1 + a_n$ — because as one row goes up by $d$ per step, the other goes down by $d$ per step, so the pair stays put.
"The number of terms, times the average of the first and last, all over two." Substitute $a_n = a_1 + (n-1)d$ if you only know the common difference rather than the last term:
$$ S_n = \frac{n\bigl(2 a_1 + (n-1)d\bigr)}{2} $$
Gauss's original problem: $a_1 = 1$, $a_{100} = 100$, $n = 100$, so $S_{100} = \frac{100 \cdot 101}{2} = 5050$.
5. Geometric sequences
Geometric sequence
A sequence in which the ratio between any two consecutive terms is the same constant $r$, called the common ratio.
Recursive: $a_{n+1} = r \cdot a_n$, with $a_1$ given.
Closed-form: $a_n = a_1 \, r^{n-1}$.
Where arithmetic sequences model linear change (something added each step), geometric sequences model multiplicative change (something scaled each step). Compound interest, population growth, radioactive decay, half-lives — all geometric.
The closed-form has the same off-by-one as the arithmetic one. The first term has had zero multiplications by $r$ applied, so the exponent of $r$ on the $n$-th term is $n-1$, not $n$.
$3, 6, 9, 12, \ldots$ — difference is constant, not ratio
$1, 2, 4, 7, 11, \ldots$ — neither
$0, 0, 0, \ldots$ from $a_1 = 0$ — the ratio is undefined
Sign of r matters
If $r$ is negative, terms alternate sign. If $|r| > 1$, terms grow in magnitude. If $|r| < 1$, terms shrink toward zero. If $r = 1$, the sequence is constant. These four regimes behave very differently when you start summing, as the next section shows.
6. Summing a geometric series
Adding up a geometric sequence uses a different — but equally clean — trick. Call the partial sum
(When $r = 1$ the sequence is just $a_1, a_1, a_1, \ldots$ and the partial sum is the boring $S_n = n a_1$.)
This "shift, line up, subtract" idea — multiply $S_n$ by the common ratio so that most terms appear in both copies and collapse on subtraction — is one of the most useful tricks in elementary algebra. The same idea reappears later in telescoping series and generating functions.
The letter $k$ underneath is the index variable; it walks from the lower bound to the upper bound, taking integer steps. The expression to the right of $\sum$ is what you plug $k$ into for each step. The two sequence sums we just derived become:
A handful of "classical" closed-form sums are worth remembering:
Sum
Closed form
Mnemonic
$\sum_{k=1}^{n} k$
$\dfrac{n(n+1)}{2}$
The triangular numbers — Gauss again
$\sum_{k=1}^{n} k^2$
$\dfrac{n(n+1)(2n+1)}{6}$
"$n$, $n+1$, $2n+1$, over six"
$\sum_{k=1}^{n} k^3$
$\left( \dfrac{n(n+1)}{2} \right)^2$
Sum of cubes is the square of the sum
Watch the bounds
$\sum_{k=1}^{n}$ has $n$ terms. $\sum_{k=0}^{n-1}$ also has $n$ terms but starts at $k = 0$. $\sum_{k=0}^{n}$ has $n + 1$ terms. A surprising number of "obvious" computations go wrong because someone counted the terms one off — always sanity-check the first and last term and the count.
8. Playground: partial sums of a geometric series
Drag the sliders to set the first term $a$ and the common ratio $r$, and the slider for $n$ to choose how many terms to add. Watch the partial sum $S_n$ approach the infinite-sum value $a / (1 - r)$ when $|r| < 1$, and run away when $|r| \geq 1$.
Sn = 2.00
a / (1 − r): 2.00 · status: converges
1.0
0.50
10
Try it
With $a = 1$ and $r = \tfrac{1}{2}$, the partial sums climb $1, 1.5, 1.75, 1.875, \ldots$ — visibly homing in on $2$ but never quite arriving. Now nudge $r$ past $1$ and the bars take off forever; nudge $r$ to $-\tfrac{1}{2}$ and they zig-zag in on $\tfrac{2}{3}$. The dashed green line is where infinity says they're going.
9. Infinite geometric series and convergence
What happens to $S_n = a_1 \dfrac{1 - r^n}{1 - r}$ when $n$ goes to infinity? Everything in that expression is fixed except $r^n$. The behaviour splits cleanly on the size of $|r|$:
If $|r| < 1$, then $r^n$ shrinks toward $0$ as $n$ grows, so $S_n$ heads toward a finite number.
If $|r| > 1$, then $|r^n|$ blows up and $S_n$ grows without bound.
If $r = 1$, every term is $a_1$ and the sum grows linearly: $n a_1$.
If $r = -1$, the partial sums oscillate $a_1, 0, a_1, 0, \ldots$ — no single value to land on.
So the infinite geometric series only "has a sum" in the first case. When $|r| < 1$, plugging $r^n \to 0$ into the formula gives:
Not approximately. Exactly. $0.\overline{9}$ and $1$ are two notations for the same real number.
Zeno's paradox dissolves
Zeno argued you can never cross a room: first you must cross half, then half the rest, then half of that, and so on forever — an infinity of steps. Each step is finite work, so surely an infinite number of finite works is infinite?
No: $\tfrac{1}{2} + \tfrac{1}{4} + \tfrac{1}{8} + \cdots = \frac{1/2}{1 - 1/2} = 1$. The infinite sum of shrinking pieces is finite. Geometric series turn out to be the most ordinary thing in the world; Zeno just didn't have the algebra.
What "converges" means, informally
An infinite series converges to a value $L$ if the partial sums $S_1, S_2, S_3, \ldots$ get and stay arbitrarily close to $L$ as $n$ grows. We've been waving at "approaches" with hand gestures here — the formal definition uses limits, which you'll meet in calculus. But the intuition is solid: pick any tolerance you like, however tiny, and after some $N$ the partial sums all lie inside that tolerance around $L$.
10. A warning: the harmonic series
It's natural to guess: "if the terms of a series shrink to zero, the sum must converge." That guess is wrong, and the smallest example that breaks it is famous enough to have a name.
The terms $1/n$ shrink to $0$, yet the sum grows without bound — it diverges.
A clean proof from Nicole Oresme (around 1350): group the terms in blocks of length $1, 2, 4, 8, \ldots$ and bound each block below by replacing every term inside with the smallest one in that block.
Every block contributes at least $\tfrac{1}{2}$. There are infinitely many blocks. So the total grows without bound — slowly, but inescapably. To get a partial sum of $20$ takes around 250 million terms, but get there it does.
The harmonic series is the canonical counterexample to the natural-but-wrong rule. The correct rule cuts the other way:
If the terms don't shrink to zero, the series definitely diverges. But shrinking to zero is not enough to guarantee convergence.
Sorting out exactly which "shrinking-to-zero" series converge and which don't is the subject matter of the convergence-tests chapter in calculus. For now, the geometric series is the safest infinite sum you can trust.
11. Common pitfalls
Off-by-one in the n-th term formula
Arithmetic: $a_n = a_1 + (n - 1)d$, not $a_1 + nd$. Geometric: $a_n = a_1 r^{n-1}$, not $a_1 r^n$. The reason is the same in both: the first term has had zero increments or multiplications applied. If you ever can't remember which exponent is right, just check $n = 1$ — the formula should reduce to $a_1$.
Confusing sequence and series
The sequence $1, \tfrac{1}{2}, \tfrac{1}{3}, \ldots$ has limit $0$. The series $1 + \tfrac{1}{2} + \tfrac{1}{3} + \cdots$ diverges. Identical-looking notation, opposite outcomes. The sequence is a list of numbers; the series is one number (or none).
Using a/(1−r) when |r| ≥ 1
The formula is only valid for $|r| < 1$. Apply it to, say, $1 + 2 + 4 + 8 + \cdots$ with $r = 2$ and you'll get $\frac{1}{1 - 2} = -1$ — a confident-looking number for a sum that is plainly infinite. Always check $|r| < 1$ before plugging in.
Starting index trips
$\displaystyle \sum_{k=0}^{\infty} a r^k = \frac{a}{1-r}$ starts at $k = 0$. The cousin $\displaystyle \sum_{k=1}^{\infty} a r^k = \frac{ar}{1-r}$ is missing the $k=0$ term and equals $r$ times as much. Read which index the problem starts from.
"Terms go to zero" is necessary, not sufficient
If $\lim_{n\to\infty} a_n \neq 0$, the series $\sum a_n$ definitely diverges. But $\lim a_n = 0$ does not guarantee convergence — the harmonic series is a permanent reminder.
12. Worked examples
Try each one before opening the solution. The goal is to recognise which formula applies and to plug in cleanly without off-by-one slip-ups.
Example 1 · Find $a_{20}$ for $5, 8, 11, \ldots$
The differences are $3$, so this is arithmetic with $a_1 = 5$ and $d = 3$.
A ball dropped from $10$ m bounces back to $\tfrac{3}{4}$ of its previous height each time. What is the total distance travelled before it comes to rest?
The initial drop is $10$ m. After that, each bounce contributes an "up" and a matching "down" of the same height. The heights form a geometric sequence with first term $10 \cdot \tfrac{3}{4} = 7.5$ and ratio $\tfrac{3}{4}$.
The treatment here borrows the layout that's now standard in algebra texts: identify the family, derive the closed-form, derive the sum, then meet the infinite case. The primary sources below are where to go when this page has done its job.
Peer-reviewed, openly licensed textbook chapter covering arithmetic and geometric sequences and series at exactly this level of rigour. The cleanest single reference for the formulas on this page.
Bite-sized videos and practice problems for both arithmetic and geometric sequences, partial sums, and recursive definitions. Best if you want to drill the mechanics until they're automatic.
Sits at the boundary between this page and the calculus convergence theory. Excellent for stepping up from "geometric series converges" to the full menagerie of convergence tests when you're ready.
Formal reference for the geometric series formula, including the generalisations (complex ratio, generating-function interpretation) that lie just outside this page's scope.
Companion article that traces the geometric-series formula from Archimedes' parabolic quadrature through its modern role as the prototype convergent series. Includes the proof that the harmonic series diverges.
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