Formula for the $n$th term of an arithmetic sequence

$u_n = u_1 + (n - 1)d$, where $d$ is the common difference.

Two formulas for the sum of the first $n$ terms of an arithmetic series

$S_n = \frac{n}{2}(2u_1 + (n - 1)d)$ or $S_n = \frac{n}{2}(u_1 + u_n)$. Use the second when the last term is known.

Given two terms $u_p$ and $u_q$ of an arithmetic sequence, how do you find $d$ quickly?

$d = \frac{u_q - u_p}{q - p}$, because $q - p$ differences separate the two terms.

Common mistake: when asked for the number of *complete* rows or items, $n = 23.3$ — what is the answer?

$23$. Round down when the question needs complete items or the total must not be exceeded; check $S_{23}$ and $S_{24}$.

SL 1.2 Arithmetic sequences and series — IB Mathematics: Analysis and Approaches notes and practice questions

Arithmetic sequences grow by the same amount each step, so their terms and sums follow two short linear formulas. Study guide, 3 practice questions with markschemes and 4 flashcards. Free.

Common difference and nth term formula · Sum of the first n terms · Sigma notation · Finding terms or n from given information · Simple interest and other linear-growth applications · Modelling approximately arithmetic data

Study guide

The big idea

An arithmetic sequence is what you get when you keep adding the same number. Think of a staircase where every step has the same height: the "height" is the common difference $d$. Because each step is identical, the $n$th term is just the first term plus $(n-1)$ steps:

\[u_n = u_1 + (n-1)d\]

Notice it is $(n-1)$, not $n$: to reach the 5th term from the 1st you take only 4 steps. Plotted against $n$, the terms lie on a straight line with gradient $d$, which is why arithmetic sequences model anything that changes by a fixed amount per period.

Adding the terms

Pair the first term with the last, the second with the second-to-last, and so on. Every pair has the same total $u_1 + u_n$, and there are $\frac{n}{2}$ pairs. That gives

\[S_n = \frac{n}{2}(u_1 + u_n) = \frac{n}{2}\big(2u_1 + (n-1)d\big).\]

Use the first version when you know the last term, the second when you know $d$.

Sigma notation

$\sum_{r=1}^{n} (3r+2)$ means "substitute $r = 1, 2, \dots, n$ and add". Expressions linear in $r$ always produce arithmetic series. Count terms carefully: $\sum_{r=4}^{10}$ has $10 - 4 + 1 = 7$ terms.

Worked example

A theatre has 18 seats in the front row and each row behind has 3 more seats than the row in front. There are 25 rows.

Seats in the back row: $u_{25} = 18 + 24 \times 3 = 90$.
Total seats: $S_{25} = \frac{25}{2}(18 + 90) = 12.5 \times 108 = 1350$.

Finding $n$. Which term of $5, 9, 13, \dots$ equals $201$? Solve $5 + (n-1)4 = 201$, so $n - 1 = 49$ and $n = 50$. If $n$ comes out as a non-integer, the value is not in the sequence.

Two unknowns. If $u_4 = 17$ and $u_{10} = 41$, subtract: $6d = 24$, so $d = 4$ and $u_1 = 17 - 12 = 5$.

Applications

Simple interest is arithmetic: USD 2000 earning 3% simple interest gains USD 60 every year, so after $n$ years the value is $2000 + 60n$. Real data that rises by roughly the same amount each period can be modelled as arithmetic by estimating $d$ from the average change, but say clearly that it is an approximation.

Common mistakes

Writing $u_n = u_1 + nd$ (one step too many).
Using $S_n$ when the question asks for a single term, or vice versa.
Miscounting terms in sigma notation that does not start at $r = 1$.
Accepting a negative or fractional $n$ when $n$ must be a positive integer.
Forgetting that $d$ can be negative for decreasing sequences.

Exam tips

With two pieces of information, write two equations in $u_1$ and $d$ and solve simultaneously (on Paper 2 the GDC can do this).
For "how many terms until the sum exceeds...", form an inequality in $n$ and use a table or solver, then check the whole-number answer.
State the formula before substituting: method marks are often given for a correct substitution even if the arithmetic slips.

Cheat sheet

Fact	Formula
$n$th term	$u_n = u_1 + (n-1)d$
Common difference	$d = u_{n+1} - u_n$
Sum (last term known)	$S_n = \frac{n}{2}(u_1 + u_n)$
Sum ($d$ known)	$S_n = \frac{n}{2}(2u_1 + (n-1)d)$
Terms in $\sum_{r=a}^{b}$	$b - a + 1$
Simple interest	value $= P + P\frac{r}{100}n$

Linear in $r$ inside $\sum$ means arithmetic.
$n$ must be a positive integer.

Key definitions

Practice questions

SLPaper 1Foundation

Question 1

An arithmetic sequence has first term $u_1 = 7$ and common difference $d = 4$.

(a) [2]

Find $u_{20}$.

Markscheme

1 Substitutes into $u_n = u_1 + (n-1)d$, e.g. $7 + 19 \times 4$
1 $u_{20} = 83$

(b) [2]

Find the sum of the first 20 terms.

Markscheme

1 Substitutes into $S_n = \frac{n}{2}(u_1 + u_n)$ or $S_n = \frac{n}{2}(2u_1 + (n-1)d)$, e.g. $10(7 + 83)$
1 $S_{20} = 900$

SLPaper 1Standard

Question 2

In an arithmetic sequence, $u_3 = 11$ and $u_8 = 31$.

(a) [3]

Find the common difference and the first term.

Markscheme

1 Forms two equations or uses $u_8 - u_3 = 5d$, e.g. $5d = 20$
1 $d = 4$
1 $u_1 = 3$

(b) [2]

Show that $S_n = 2n^2 + n$.

Markscheme

1 Substitutes into $S_n = \frac{n}{2}(2u_1 + (n-1)d)$, e.g. $\frac{n}{2}(6 + 4(n-1))$
1 Simplifies correctly, e.g. $\frac{n}{2}(4n + 2) = 2n^2 + n$

(c) [2]

Find the smallest value of $n$ for which $S_n > 500$.

Markscheme

1 Tests values either side, e.g. $S_{15} = 465$ and $S_{16} = 528$
1 $n = 16$

SLPaper 2Challenging

Question 3

A concert hall is being designed with curved rows of seats. The first row has 18 seats and each row after that has 3 more seats than the row in front of it.

(a) [2]

Find the number of seats in row 15.

Markscheme

1 Substitutes into $u_n = u_1 + (n-1)d$, e.g. $18 + 14 \times 3$
1 $60$ seats

(b) [4]

The hall can hold at most 1200 seats. Find the greatest number of complete rows that can be built.

Markscheme

1 Uses sum formula, e.g. $\frac{n}{2}(36 + 3(n-1)) \le 1200$
1 Correct quadratic, e.g. $3n^2 + 33n - 2400 \le 0$ or $n^2 + 11n - 800 \le 0$
1 Solves to get $n = 23.3$ (or finds $S_{23} = 1173$ and $S_{24} = 1260$)
1 $23$ complete rows

(c) [2]

Find how many of the 1200 seats are left over after these complete rows are built.

Markscheme

1 $S_{23} = 1173$
1 $27$ seats left over

Fact	Formula
\(n\)th term	\(u_n = u_1 + (n-1)d\)
Common difference	\(d = u_{n+1} - u_n\)
Sum (last term known)	\(S_n = \frac{n}{2}(u_1 + u_n)\)
Sum (\(d\) known)	\(S_n = \frac{n}{2}(2u_1 + (n-1)d)\)
Terms in \(\sum_{r=a}^{b}\)	\(b - a + 1\)
Simple interest	value \(= P + P\frac{r}{100}n\)

SL 1.2 Arithmetic sequences and series

Study guide

The big idea

Adding the terms

Sigma notation

Worked example

Applications

Common mistakes

Exam tips

Cheat sheet

Key definitions

Practice questions

Question 1

Question 2

Question 3

Key facts