Sequences and Series

A sequence is simply an ordered list of numbers that follows a rule, and a series is what you get when you add those numbers up. That sounds modest, but this one idea underlies compound interest, computer algorithms, the way a bouncing ball eventually comes to rest, and even the number $\pi$. Once you can spot the pattern in a list and turn it into a formula, you stop adding numbers one painful term at a time and start reasoning about infinitely many of them at once.

This page will teach you to recognize the two most important families — arithmetic and geometric — to write sums compactly with sigma notation, to compute partial sums with formulas instead of a calculator, and to understand the beautiful, counterintuitive fact that an infinite list of positive numbers can add up to a finite total.

Learning Objectives

• Distinguish a sequence from a series, and a term from a sum.
• Identify arithmetic sequences (constant difference) and geometric sequences (constant ratio) and write their $n$th-term formulas.
• Read and write sums using sigma ($\Sigma$) notation.
• Compute partial sums of arithmetic and geometric series with closed-form formulas.
• Determine when an infinite geometric series converges, and find its sum.
• Apply these tools to real problems in finance, physics, and computing.

Quick Answer

A sequence lists terms; a series adds them. An arithmetic sequence adds a fixed number $d$ each step, so its $n$th term is $a_n = a_1 + (n-1)d$. A geometric sequence multiplies by a fixed ratio $r$ each step, so $a_n = a_1 r^{\,n-1}$. The sum of the first $n$ terms is $S_n = \frac{n}{2}(a_1 + a_n)$ for arithmetic series and $S_n = a_1\frac{1 - r^n}{1 - r}$ for geometric series. An infinite geometric series converges only when $|r| < 1$, and then its sum is $S = \frac{a_1}{1 - r}$. Sigma notation, written $\sum_{k=1}^{n} a_k$, is just shorthand for "add these terms up."

Where It Came From

People needed to add long, patterned lists long before they had formulas, and the history of this topic is really the history of clever shortcuts.

The most famous story belongs to Carl Friedrich Gauss. As a schoolboy around 1785, he was reportedly told (as busywork) to add every whole number from 1 to 100. Within moments he wrote the answer: 5050. He had noticed that if you pair the first and last terms ($1 + 100 = 101$), the second and second-to-last ($2 + 99 = 101$), and so on, you get 50 pairs each summing to 101, giving $50 \times 101 = 5050$. That single insight — pairing terms from the ends inward — is the arithmetic series formula, and it works for any arithmetic list, not just 1 to 100. The need it answered was ancient and practical: surveyors, accountants, and astronomers constantly faced sums of evenly spaced quantities.

Infinite series carry an even older and more philosophical origin: the paradoxes of Zeno of Elea (around 450 BCE). Zeno argued that motion is impossible. To walk across a room you must first cover half the distance, then half of what remains, then half of that, forever — infinitely many steps, so (he claimed) you can never arrive. The resolution, made rigorous only two thousand years later, is that the infinite sum $\tfrac12 + \tfrac14 + \tfrac18 + \cdots$ adds up to exactly 1. Infinitely many pieces can have a finite total. Mathematicians from Archimedes (who summed geometric series to find areas) through the calculus of Newton, Leibniz, and finally the rigorous limit definitions of Cauchy in the 1820s turned Zeno's puzzle into the modern theory of convergence.

Arithmetic Sequences and Series

An arithmetic sequence changes by the same amount, the common difference $d$, at every step: $3, 7, 11, 15, 19, \dots$ has $d = 4$. Because each term is the first term plus some number of steps of size $d$:

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

To sum the first $n$ terms, use Gauss's pairing trick. Writing the sum forwards and backwards and adding gives $2S_n = n(a_1 + a_n)$, so:

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

Worked example. Find the 20th term and the sum of the first 20 terms of $3, 7, 11, 15, \dots$

Here $a_1 = 3$ and $d = 4$. The 20th term is

$a_{20} = 3 + (20-1)\times 4 = 3 + 76 = 79.$

The sum is

$S_{20} = \frac{20}{2}\,(3 + 79) = 10 \times 82 = 820.$

Check Gauss's own case with this formula: $a_1 = 1$, $a_{100} = 100$, so $S_{100} = \frac{100}{2}(1 + 100) = 50 \times 101 = 5050$. It matches.

Geometric Sequences and Series

A geometric sequence multiplies by the same common ratio $r$ at each step: $2, 6, 18, 54, \dots$ has $r = 3$. The $n$th term is

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

For the sum, there is an elegant trick. Write $S_n = a_1 + a_1 r + \cdots + a_1 r^{\,n-1}$, multiply the whole thing by $r$, and subtract. Almost every term cancels, leaving:

$S_n = a_1\,\frac{1 - r^{\,n}}{1 - r}\qquad (r \neq 1).$

Worked example. Sum the first 6 terms of $2, 6, 18, 54, \dots$

Here $a_1 = 2$, $r = 3$, $n = 6$:

$S_6 = 2 \times \frac{1 - 3^{6}}{1 - 3} = 2 \times \frac{1 - 729}{-2} = 2 \times \frac{-728}{-2} = 2 \times 364 = 728.$

Verify by hand: $2 + 6 + 18 + 54 + 162 + 486 = 728$. Exactly right.

Notice how geometric growth explodes: six terms already exceed 700, while an arithmetic sequence with the same start would still be small. This gap is why geometric reasoning governs compound interest, population growth, and viral spread.

Sigma Notation

Writing out long sums is tedious and error-prone, so mathematicians use the capital Greek letter sigma, $\Sigma$, as a compact instruction to add:

$\sum_{k=1}^{n} a_k = a_1 + a_2 + \cdots + a_n.$

The letter $k$ is the index, it starts at the number below the sigma and increases by 1 up to the number on top. For example:

$\sum_{k=1}^{5} (2k + 1) = 3 + 5 + 7 + 9 + 11 = 35.$

Here each term is generated by plugging $k = 1, 2, 3, 4, 5$ into the rule $2k + 1$. This is an arithmetic series with $a_1 = 3$, $a_5 = 11$, so the formula confirms $\frac{5}{2}(3 + 11) = \frac{5}{2}\times 14 = 35$.

Two facts make sigma notation powerful: you can pull constants out, $\sum c\,a_k = c\sum a_k$, and split sums, $\sum (a_k + b_k) = \sum a_k + \sum b_k$. These let you rewrite messy sums into standard forms you already know how to compute.

Infinite Geometric Series and Convergence

What happens if a geometric series never stops? Look again at $r^{\,n}$ in the sum formula. If $|r| < 1$, then $r^{\,n}$ shrinks toward 0 as $n$ grows without bound. Dropping that vanishing term leaves a finite total:

$S = \frac{a_1}{1 - r}\qquad \text{valid only when } |r| < 1.$

If $|r| \geq 1$ the terms do not shrink, the partial sums grow forever, and the series diverges (no finite sum).

Worked example — Zeno resolved. The distances $\tfrac12, \tfrac14, \tfrac18, \dots$ form a geometric series with $a_1 = \tfrac12$ and $r = \tfrac12$. Since $|r| < 1$:

$S = \frac{\tfrac12}{1 - \tfrac12} = \frac{\tfrac12}{\tfrac12} = 1.$

The infinitely many steps add to exactly 1 — you do cross the room. Zeno's paradox dissolves the moment you accept that an unending sum can converge.

Worked example — repeating decimals. The decimal $0.4444\dots$ equals $\tfrac{4}{10} + \tfrac{4}{100} + \tfrac{4}{1000} + \cdots$, a geometric series with $a_1 = \tfrac{4}{10}$ and $r = \tfrac{1}{10}$:

$S = \frac{4/10}{1 - 1/10} = \frac{4/10}{9/10} = \frac{4}{9}.$

So $0.\overline{4} = \tfrac{4}{9}$ — a satisfying proof that repeating decimals are always fractions.

Real-World Applications

• Compound interest and savings. Depositing a fixed amount each year that then earns interest produces a geometric series; the "future value of an annuity" formula is exactly $S_n = a_1\frac{1 - r^n}{1 - r}$ in disguise.
• Loan and mortgage payments. Amortization schedules sum geometric series to figure out the level monthly payment that pays off a balance.
• Physics of damped motion. A bouncing ball that rebounds to a fixed fraction of its previous height travels a total distance found by an infinite geometric series.
• Computer science. Analyzing algorithms (like the cost of doubling an array, or a binary search's halving) relies on geometric-series sums; so does estimating the total work in recursive processes.
• Medicine. Drug dosing at regular intervals reaches a steady-state concentration modeled by a convergent geometric series.

Common Mistakes

1. Confusing a sequence with a series. A sequence is the list ($2, 4, 6, \dots$); a series is the sum ($2 + 4 + 6 + \cdots$). Why it matters: the questions "what is the 10th term?" and "what is the sum of 10 terms?" have completely different answers. Correction: decide first whether you are locating a term ($a_n$) or totaling terms ($S_n$).

2. Applying the infinite-sum formula when $|r| \geq 1$. Students plug into $\frac{a_1}{1-r}$ even for, say, $r = 2$, getting a negative "sum" for a series of growing positive terms. Why it's wrong: the series diverges, so no finite sum exists; the formula is meaningless there. Correction: always check $|r| < 1$ before using $S = \frac{a_1}{1-r}$.

3. Off-by-one errors in $a_n = a_1 + (n-1)d$. People multiply $d$ by $n$ instead of $n-1$. Why it's wrong: the first term needs zero steps of $d$, the $n$th term needs $n-1$ steps. Correction: test on term 1 — the formula must return $a_1$, which it does only with $(n-1)$.

Comparison and Connections

Arithmetic and geometric progressions are the two building blocks; the key contrast is addition versus multiplication.

Feature	Arithmetic	Geometric
Step rule	add $d$	multiply by $r$
$n$th term	$a_1 + (n-1)d$	$a_1 r^{\,n-1}$
Partial sum $S_n$	$\frac{n}{2}(a_1 + a_n)$	$a_1\frac{1-r^n}{1-r}$
Infinite sum	never converges (unless all terms 0)	converges to $\frac{a_1}{1-r}$ if $\vert r\vert < 1$
Growth shape	straight line	exponential curve

Arithmetic sequences are the discrete cousins of linear functions; geometric sequences are the discrete cousins of exponential functions. Infinite series are the gateway to calculus, where limits, Taylor series, and integrals all rest on summing infinitely many pieces.

Practice Questions

Recall

State the $n$th-term formula for an arithmetic sequence and for a geometric sequence. Answer: Arithmetic: $a_n = a_1 + (n-1)d$. Geometric: $a_n = a_1 r^{\,n-1}$.

Understanding

Explain why the infinite series $1 + 2 + 4 + 8 + \cdots$ has no finite sum, while $1 + \tfrac12 + \tfrac14 + \cdots$ does. Guidance: The first has $r = 2$ so $|r| \geq 1$; terms grow and partial sums increase without bound (diverges). The second has $r = \tfrac12$ so $|r| < 1$; terms shrink to 0 and the sum converges to $\frac{1}{1 - 1/2} = 2$.

Application

A theater has 18 seats in the front row and 3 more in each successive row, for 25 rows. How many seats total? Answer: Arithmetic with $a_1 = 18$, $d = 3$, $n = 25$. Then $a_{25} = 18 + 24\times 3 = 90$, and $S_{25} = \frac{25}{2}(18 + 90) = \frac{25}{2}\times 108 = 1350$ seats.

Analysis

A ball is dropped from 10 m and each bounce reaches 60% of the previous height. Find the total vertical distance it travels before coming to rest. Guidance: It falls 10 m, then each bounce goes up and back down. Up-distances form a geometric series $6 + 3.6 + \cdots$ with $a_1 = 6$, $r = 0.6$, summing to $\frac{6}{1 - 0.6} = 15$; the down-distances after the first drop equal the same 15. Total $= 10 + 15 + 15 = 40$ m.

FAQ

Is zero a valid common ratio? No — if $r = 0$ every term after the first is 0, which is trivial, and the formulas assume $r \neq 0$ (and $r \neq 1$ for the partial-sum formula, since that would divide by zero).

Can a common difference or ratio be negative? Yes. A negative $d$ gives a decreasing arithmetic sequence ($10, 7, 4, \dots$). A negative $r$ gives an alternating geometric sequence ($3, -6, 12, \dots$); it still converges if $|r| < 1$, e.g. $r = -\tfrac12$.

What's the difference between $S_n$ and $S$? $S_n$ is a partial sum — the total of the first $n$ terms (always finite). $S$ (no subscript) is the infinite sum, the limit of $S_n$ as $n \to \infty$, which only exists for convergent series.

Why does the index letter not matter? In $\sum_{k=1}^{n} a_k$ the $k$ is a "dummy variable" — using $i$ or $j$ instead gives the identical sum. It's just a counter that disappears once the addition is done.

How is this connected to calculus? Calculus is built on infinite sums. Integrals are limits of sums of infinitely many thin slices, and functions like $e^x$ and $\sin x$ are defined by infinite series. Mastering convergence here is direct preparation.

Quick Revision

• Sequence = ordered list; series = sum of the list.
• Arithmetic: constant difference $d$; $a_n = a_1 + (n-1)d$; $S_n = \frac{n}{2}(a_1 + a_n)$.
• Geometric: constant ratio $r$; $a_n = a_1 r^{\,n-1}$; $S_n = a_1\frac{1 - r^n}{1 - r}$.
• Infinite geometric sum: $S = \frac{a_1}{1 - r}$, only if $|r| < 1$; otherwise it diverges.
• Sigma notation: $\sum_{k=1}^{n} a_k$ means add terms from $k=1$ to $k=n$.
• Gauss's trick: pair ends inward. Zeno's series $\tfrac12 + \tfrac14 + \cdots = 1$.

Related Topics

Prerequisites

• Algebra overview

Related Topics

• Exponential and logarithmic functions (geometric growth in continuous form)
• Linear functions (the continuous analog of arithmetic sequences)

Next Topics

• Limits and convergence in Calculus
• Compound interest in Financial Mathematics