Arithmetic Sequences and Series: Formulas, Sums and Examples

What is an arithmetic sequence?

An arithmetic sequence (or arithmetic progression) is a list of numbers a_1,a_2,a_3,… in which the difference between each term and the one before it is always the same number d, called the common difference. The notation is a_n for the term in position n, and in many school books U_n.

That gives the rule a_n+1=a_n+d: to get the next term, add d. A sequence is the list itself; a series is what you get by adding its terms, which the sum section below takes up.

How to tell. Subtract each term from the next, always later minus earlier. If every difference is the same, the sequence is arithmetic. One difference that breaks the pattern is enough to rule it out. A sequence whose terms are multiplied by the same number each time is a geometric sequence, built from the powers in the article on exponents.

Two useful facts follow at once. Each term is the average of its two neighbors, a_n=(a_n-1+a_n+1)/(2), which is why the word arithmetic is used: the arithmetic mean sits in the middle. And differences may be negative or fractions, so the integers rules for negative numbers and the rational numbers rules for fractions apply.

Where they appear. Equal steps are everywhere: the heights of the steps in a staircase, seats in rows that each have 3 more than the one in front, a monthly saving that rises by a fixed amount, a taxi fare that adds a fixed charge per kilometer, a simple (not compound) interest balance. Type a list of numbers below and see whether it is arithmetic.

How do you find the n-th term of an arithmetic sequence?

The n-th term of an arithmetic sequence is a_n=a_1+(n-1)d: start at the first term and add the difference n-1 times. The reason for n-1 is that going from term 1 to term n takes n-1 steps, not n.

For 5,9,13,… the first term is 5 and d=4, so a_n=5+(n-1)·4=4n+1 and a_20=5+19·4=81.

It is a linear function. Expanding gives a_n=dn+(a_1-d), which is a straight line in n with slope d and, for n=0, the intercept a_1-d. That is why the points of an arithmetic sequence lie on a line, and why the section on algebraic expressions calls it linear: it has degree 1. A positive d tilts the line up, a negative d tilts it down, and d=0 makes it flat.

How many terms? Solve a_n=ℓ for n: n=(ℓ-a_1)/(d)+1. For 7,12,17,…,102 this is (102-7)/(5)+1=20. The answer must be a positive whole number, which also tells you whether a number belongs to the sequence: for 100 it is (100-7)/(5)+1=19.6, so 100 is not a term. The same count gives the number of multiples of 7 from 100 to 500: the first is 105=7·15 and the last 497=7·71, so n=71-15+1=57.

Choose a first term, a difference and a length below, and watch the terms, the formula and the straight line.

How do you find a sequence from two of its terms?

Two terms and their positions fix an arithmetic sequence completely: subtract them to find d, then step back to a_1. Between term p and term q there are q-p steps of size d, so a_q-a_p=(q-p)d and

d=(a_q-a_p)/(q-p), a_1=a_p-(p-1)d.

Example: a_3=11 and a_7=23. Then d=(23-11)/(7-3)=(12)/(4)=3 and a_1=11-2·3=5, so the sequence is 5,8,11,14,… with a_n=3n+2. Check: a_3=11, a_7=23.

The usual slip is to divide by the wrong number: divide by the number of steps q-p (here 4), not by p, q or q+p. The two positions can be given in either order, because swapping them changes both the top and the bottom signs.

Arithmetic means. Inserting k numbers between a and b so that the whole list is arithmetic is the same problem with q-p=k+1 steps: d=(b-a)/(k+1). Three means between 4 and 24 have d=(20)/(4)=5 and are 9, 14, 19. With one mean the answer is the plain average (a+b)/(2).

Give any two terms below and see the whole working.

How do you find the sum of an arithmetic series?

The sum of the first n terms of an arithmetic sequence is S_n=\dfrac n2(a_1+a_n)=\dfrac n2\bigl(2a_1+(n-1)d\bigr): the number of terms times the average of the first and the last. A sum of terms like this is called an arithmetic series, and S_n is its n-th partial sum.

Gauss's trick. Write the sum twice, once forward and once backward, and add the columns:

S_n=a_1+a_2+…+a_n

S_n=a_n+a_n-1+…+a_1

In every column the sum is the same, a_1+a_n, because moving one place along adds d to one term and takes it from the other. There are n columns, so 2S_n=n(a_1+a_n). The story goes that Gauss, as a schoolboy, found 1+2+…+100 this way: 50 pairs of 101, which is 5050.

Examples.

• 5+9+13+… with 20 terms: a_20=81, so S_20=(20)/(2)(5+81)=860.
• 3+8+13+…+98: it has (98-3)/(5)+1=20 terms, so the sum is (20)/(2)(3+98)=1010.
• The multiples of 7 from 105 to 497, 57 of them: (57)/(2)(105+497)=17,157.

Three sums worth knowing. 1+2+…+n=(n(n+1))/(2) (the triangular numbers 1, 3, 6, 10, 15); the first n odd numbers add to n^2, since 1+3+5+…+(2n-1)=\frac n2·2n; and the first n even numbers add to n(n+1).

Sigma notation writes a series compactly: \sum_k=1^n(a_1+(k-1)d) means "add the terms for k=1 up to k=n". For example \sum_k=1^10(3k+2)=5+8+…+32=(10)/(2)(5+32)=185.

S_n is quadratic in n. Expanding gives S_n=\frac d2n^2+\bigl(a_1-\frac d2\bigr)n, a quadratic with no constant term, which is how to recognize one. And you can run it backward: a_n=S_n-S_n-1 for n\ge2 (and a_1=S_1). If S_n=2n^2+3n then a_1=5 and a_n=\bigl(2n^2+3n\bigr)-\bigl(2(n-1)^2+3(n-1)\bigr)=4n+1, which is the sequence 5,9,13,… again.

Try the pairing for your own sequence below.

What happens to an infinite arithmetic series?

An infinite arithmetic series has no finite sum unless every term is 0: its partial sums grow without bound. Adding more terms keeps changing the total by a non-zero amount, and with d≠0 the terms themselves get larger in size, so the sum cannot settle.

• If d>0 the partial sums S_n=\frac d2n^2+\bigl(a_1-\frac d2\bigr)n eventually grow to +∞.
• If d<0 they eventually fall to -∞.
• If d=0 and a_1≠0 then S_n=na_1 also grows without bound.

Such a series is said to diverge. This is a sharp contrast with a geometric series with a common ratio of size less than 1, such as \frac12+\frac14+\frac18+…, whose terms shrink fast enough for the sum to settle at 1. In practice that means every question about an arithmetic series asks for a finite number of terms. If a problem seems to ask for the sum of an infinite arithmetic series, it is a geometric series in disguise or the answer is "it diverges".

How do you solve word problems with arithmetic sequences?

Read the problem for three things, the first term a_1, the common difference d and what is asked (a term, a count, or a sum), then use the matching formula. Check at the end that the answer is sensible and a whole number where it counts objects.

Notice that "each row has 3 more" and "50 more each month" give d directly, and that a drop gives a negative d. Notice too that the term in position 15 is only 14 steps from the first, the same n-1 as in the formula.

Simple interest is an arithmetic sequence: a deposit of P earning a fixed amount I a year is worth P, P+I, P+2I,…, so after n years it is P+nI. Compound interest multiplies by a fixed ratio instead and gives a geometric sequence.

How do you work with arithmetic sequences in Python and JavaScript?

In Python range(first, stop, step) is an arithmetic sequence of integers, and sum of it is an arithmetic series; in JavaScript you build the terms with Array.from. The stop value of range is excluded, which is the usual source of off-by-one mistakes.

>>> list(range(3, 40, 4))                 # first, stop (excluded), step
[3, 7, 11, 15, 19, 23, 27, 31, 35, 39]
>>> a1, d, n = 3, 4, 10
>>> a1 + (n - 1) * d                         # the n-th term
39
>>> n * (2 * a1 + (n - 1) * d) // 2          # the sum of the first n terms (exact)
210
>>> sum(range(1, 101))                       # Gauss's sum
5050
>>> from itertools import accumulate
>>> list(accumulate(range(1, 6)))            # partial sums: 1, 3, 6, 10, 15
[1, 3, 6, 10, 15]
>>> len(range(105, 498, 7))                  # multiples of 7 from 105 to 497
57
>>> sum(range(105, 498, 7))
17157
>>> [i * 0.1 for i in range(4)]              # a float step drifts
[0.0, 0.1, 0.2, 0.30000000000000004]

const terms = (a1, d, n) => Array.from({ length: n }, (_, i) => a1 + i * d)
terms(3, 4, 5)                               // [3, 7, 11, 15, 19]
terms(3, 4, 5).reduce((s, x) => s + x, 0)    // 55
(5 * (2 * 3 + 4 * 4)) / 2                    // 55, from S = n(2a + (n-1)d)/2
Array.from({ length: 4 }, (_, i) => i * 0.1) // [0, 0.1, 0.2, 0.30000000000000004]

#include <iostream>
#include <cstdio>
using namespace std;

int main() {
  int a1 = 3, d = 4, n = 10;
  for (int i = 0; i < n; i++) cout << a1 + i * d << " ";   // the first ten terms
  cout << endl;
  cout << a1 + (n - 1) * d << endl;                        // the n-th term
  cout << n * (2 * a1 + (n - 1) * d) / 2 << endl;          // the sum of n terms, exactly
  int gauss = 0;
  for (int i = 1; i <= 100; i++) gauss += i;
  cout << gauss << endl;                                   // Gauss's sum
  int count = 0, total = 0;
  for (int m = 105; m <= 497; m += 7) { count++; total += m; }
  cout << count << " " << total << endl;                   // multiples of 7 from 105 to 497
  for (int i = 0; i < 4; i++) printf("%.17f ", i * 0.1);   // a float step drifts
  cout << endl;
  return 0;
}

/* output:
3 7 11 15 19 23 27 31 35 39
39
210
5050
57 17157
0.00000000000000000 0.10000000000000001 0.20000000000000001 0.30000000000000004
*/

The pitfalls, in order of how often they bite:

For exact fractional steps use fractions.Fraction, as in the article on rational numbers. Check a formula by comparing it with sum(range(...)) on small cases before trusting it on large ones.

Where do arithmetic sequences come from?

Arithmetic progressions are among the oldest patterns in mathematics, long before there was a notation for them.

• c. 1550 BCE. The Rhind Papyrus has a problem that shares 10 measures of barley among 10 men so that each gets the same amount more than the one before: an arithmetic progression with difference \frac18.
• 5th century CE. The Chinese Zhang Qiujian suanjing has a woman weaving cloth who weaves a fixed extra amount each day; the calculation was done on counting rods.
• 499 CE. Aryabhata gave rules in his Aryabhatiya for the sum of an arithmetic progression and for the number of its terms.
• About 1787. The classroom story about Gauss, aged nine or ten, is that his teacher set 1+2+…+100 to keep the class busy and Gauss answered 5050 at once, by pairing 1 with 100, 2 with 99 and so on. The story was written down long afterward and the details vary, but the idea of pairing is certainly his kind of insight, and it is the proof still given today.

The word arithmetic goes back to the Greek arithmos, number.

What are the common mistakes with arithmetic sequences?

The most common mistakes with arithmetic sequences are the nine below, each with the correct statement and a number that shows why.

Practice: test your understanding

These questions mix everything above. A wrong answer costs nothing here: read the hint and try again.

Summary: arithmetic sequences and series at a glance

• Definition: a_n+1-a_n=d is constant; each term is the average of its neighbors.
• n-th term: a_n=a_1+(n-1)d=dn+(a_1-d), a straight line in n.
• Number of terms: n=(ℓ-a_1)/(d)+1, which must be a positive whole number.
• From two terms: d=(a_q-a_p)/(q-p), a_1=a_p-(p-1)d; k means between a and b have d=(b-a)/(k+1).
• Sum: S_n=\frac n2(a_1+a_n)=\frac n2\bigl(2a_1+(n-1)d\bigr), quadratic in n; a_n=S_n-S_n-1.
• Special sums: 1+…+n=\fracn(n+1)2; the first n odd numbers give n^2.
• Infinite series: an arithmetic series diverges.
• Code: range(first, stop, step) with the stop excluded; use integers or Fraction, not repeated float steps.

Glossary: key terms

Sequence

An ordered list of numbers, called terms, written a1, a2, a3 and so on.

Term

One number in a sequence; the n-th term is written a sub n.

Arithmetic sequence

A sequence in which the difference between each term and the one before it is always the same, such as 3, 7, 11, 15.

Common difference

The constant amount d added to each term of an arithmetic sequence to get the next; it can be positive, negative, zero or a fraction.

Recursive formula

A rule that gives each term from the one before it, such as a sub n plus 1 equals a sub n plus d.

Explicit formula

A rule that gives the n-th term directly from n, such as a sub n equals a sub 1 plus n minus 1 times d.

Series

The sum of the terms of a sequence, such as 3 plus 7 plus 11 plus 15.

Partial sum

The sum S sub n of the first n terms of a sequence.

Sigma notation

A compact way to write a sum using the Greek letter sigma, with the first and last values of the index written below and above it.

Arithmetic mean

The average of two numbers, half their sum; each term of an arithmetic sequence is the arithmetic mean of its neighbors.

Triangular number

A sum 1 plus 2 plus up to n, equal to n times n plus 1 over 2, such as 1, 3, 6, 10 and 15.

Divergent series

A series whose partial sums do not settle on a finite total; every infinite arithmetic series with a non-zero term is divergent.

Geometric sequence

A sequence in which each term is the previous one times the same fixed number, such as 2, 4, 8, 16.

Frequently asked questions

References

1. James Stewart, Lothar Redlin and Saleem Watson (2016). Precalculus: Mathematics for Calculus (7th ed.), chapter 12 (sequences and series). Cengage Learning.
2. Arnold Buffum Chace (1927). The Rhind Mathematical Papyrus. Mathematical Association of America.
3. Shen Kangshen, John Crossley and Anthony Lun (1999). The Nine Chapters on the Mathematical Art: Companion and Commentary. Oxford University Press.
4. Walter Eugene Clark (1930). The Aryabhatiya of Aryabhata: An Ancient Indian Work on Mathematics and Astronomy (Ganita section). University of Chicago Press.
5. Wolfgang Sartorius von Waltershausen (1856). Gauss zum Gedächtniss. S. Hirzel.
6. Ronald Graham, Donald Knuth and Oren Patashnik (1994). Concrete Mathematics: A Foundation for Computer Science (2nd ed.), chapter 2 (sums). Addison-Wesley.
7. Python Software Foundation. The Python Standard Library: range, sum and itertools.accumulate. docs.python.org. https://docs.python.org/3/library/stdtypes.html#range.

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