Arithmetic and geometric sequences and series

IB Mathematics AI HL· 5 min read

1. Arithmetic Sequences★★☆☆☆⏱ 15 min

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📘 Definition

Arithmetic Sequence

$u_{n}$ = nth term, $d$ = common difference

A sequence where consecutive terms differ by a constant value called the common difference.

Example:

3, 7, 11, 15, ... has $d = 4$

The general formula for the nth term of an arithmetic sequence is derived from the linear pattern of increasing or decreasing terms:

u_n = u_1 + (n-1)d

📐 Worked Example

The 5th term of an arithmetic sequence is 22, and the 10th term is 47. Find the 15th term.

1
Set up simultaneous equations using the nth term formula:
2
$\begin{align*} u_1 + 4d &= 22 \\ u_1 + 9d &= 47 \end{align*}$
3
Subtract the first equation from the second to eliminate $u_{1}$ :
4
$5d = 25 \implies d = 5$
5
Substitute $d = 5$ back to find the first term:
6
$u_1 + 4(5) = 22 \implies u_1 = 2$
7
Calculate the 15th term:
8
$u_{15} = 2 + (15-1)(5) = 72$

Exam tip:

Always remember the number of differences between term 1 and term n is $(n - 1)$ , not $n$ .

2. Arithmetic Series★★☆☆☆⏱ 15 min

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📘 Definition

Arithmetic Series

$S_{n}$ = sum of first n terms

The sum of the first n terms of an arithmetic sequence.

Example:

Sum of first 4 terms of 3, 7, 11, 15 is $3 + 7 + 11 + 15 = 36$

The sum formula is derived by pairing the first and last term, which always add to the same constant value. There are two equivalent forms:

S_n = \frac{n}{2}\left(u_1 + u_n\right) = \frac{n}{2}\left(2u_1 + (n-1)d\right)

📐 Worked Example

Find the sum of the first 30 terms of the sequence $7, 4, 1, - 2, ...$

1
Identify values: $u_{1} = 7$ , $d = - 3$ , $n = 30$
2
Substitute into the sum formula:
3
$S_{30} = \frac{30}{2}\left(2(7) + (30-1)(-3)\right)$
4
Simplify to get the final result:
5
$15\left(14 - 87\right) = 15(-73) = -1095$

3. Geometric Sequences★★★☆☆⏱ 20 min

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📘 Definition

Geometric Sequence

$u_{n}$ = nth term, $r$ = common ratio

A sequence where each consecutive term is multiplied by a constant common ratio $r$ .

Example:

2, 6, 18, 54, ... has $r = 3$

Repeated multiplication by the common ratio gives the exponential nth term formula for geometric sequences:

u_n = u_1 r^{n-1}

📐 Worked Example

A geometric sequence has a third term of 12 and a sixth term of 96. Find the first term.

1
Write equations for both given terms:
2
$\begin{align*} u_1 r^2 &= 12 \\ u_1 r^5 &= 96 \end{align*}$
3
Divide the second equation by the first to eliminate $u_{1}$ :
4
$r^3 = \frac{96}{12} = 8 \implies r = 2$
5
Substitute $r = 2$ back to solve for $u_{1}$ :
6
$u_1 (2)^2 = 12 \implies 4u_1 = 12 \implies u_1 = 3$

Exam tip:

Common ratio can be negative, leading to alternating positive and negative terms.

4. Geometric Series (Finite and Infinite)★★★☆☆⏱ 20 min

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📘 Definition

Geometric Series

$S_{n}$ = sum of first n terms, $S_{\infty}$ = infinite sum

The sum of the first n terms of a geometric sequence. An infinite sum exists only for convergent series.

The sum of a finite geometric series is given by:

S_n = \frac{u_1(1 - r^n)}{1 - r}, \quad r \neq 1

An infinite geometric series converges to a finite sum only if the absolute value of the common ratio is less than 1. When this condition is met, the infinite sum is:

S_\infty = \frac{u_1}{1 - r}, \quad |r| < 1

📐 Worked Example

Find the sum to infinity of the series $18 + 6 + 2 + \frac{2}{3} + ...$

1
Identify $u_{1} = 18$ , calculate common ratio $r = \frac{6}{18} = \frac{1}{3}$
2
Check convergence: $∣ \frac{1}{3} ∣ < 1$ , so infinite sum exists
3
Substitute into the infinite sum formula:
4
$S_\infty = \frac{18}{1 - \frac{1}{3}} = \frac{18}{\frac{2}{3}} = 27$

5. Common Pitfalls

Wrong move:

Using $n$ instead of $n - 1$ in nth term formulas

Why:

The first term occurs at n=1, so the exponent or difference is always one less than the term number

Correct move:

Always write $(n - 1)$ for the exponent in geometric nth terms and for the difference in arithmetic nth terms

Wrong move:

Skipping the convergence check for infinite geometric series

Why:

The infinite sum formula only works for $∣ r ∣ < 1$ , you will lose marks for not confirming convergence

Correct move:

Always state that $∣ r ∣ < 1$ before using the infinite sum formula

Wrong move:

Mixing up arithmetic and geometric sum formulas

Why:

The formulas are structurally different, and using the wrong one leads to zero marks for the question

Correct move:

Remember: arithmetic is linear (uses $d$ for difference), geometric is exponential (uses $r$ for ratio)

Wrong move:

Sign errors with negative common ratios

Why:

Raising a negative number to an even or odd power changes the sign of $r^{n}$ , leading to incorrect sum calculations

Correct move:

Evaluate $r^{n}$ separately, check the sign before substituting into the sum formula

Wrong move:

Counting the wrong number of terms between two given terms

Why:

Some students add an extra 1 when calculating the difference in term positions

Correct move:

The difference in exponents/positions is always $n_{2} - n_{1}$ for two given terms

6. Quick Reference Cheatsheet

Property	Arithmetic	Geometric
nth term	$u_{n} = u_{1} + (n - 1) d$	$u_{n} = u_{1} r^{n - 1}$
Common difference/ratio	$d = u_{n + 1} - u_{n}$	$r = \frac{u _{n + 1}}{u _{n}}$
Sum of first $n$ terms	$S_{n} = \frac{n}{2} (2 u_{1} + (n - 1) d)$	$S_{n} = \frac{u _{1} ( 1 - r ^{n} )}{1 - r}, r \neq = 1$
Infinite sum	No finite sum (unless $d = 0$ )	$S_{\infty} = \frac{u _{1}}{1 - r}, ∣ r ∣ < 1$

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2021 · 1
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What's Next

This subtopic is a core foundation for many key topics in IB AI HL, including financial mathematics (compound interest is a geometric sequence application), exponential growth and decay models, and recurrence relation modelling. Sequences and series also frequently combine with logarithms, as solving for the number of terms $n$ in a geometric sequence requires logarithmic manipulation. Mastering these formulas and problem-solving patterns will prepare you for the modelling-focused questions that make up a large portion of IB AI HL exams.