Sequences and Recurrence Relations for the TMUA

Updated August 2025

An exploration of sequences defined by explicit formulae and recurrence relations. This guide covers how to generate terms, spot emerging patterns, and calculate sums using sigma notation, while avoiding common pitfalls like the fence post error in the Test of Mathematics for University Admission.

Core concept

A sequence is an ordered list of numbers where each term is determined by its position $n$ or by a recurrence relation $x_{n+1} = f(x_n)$ that calculates the next term from the previous one.

Understanding Sequences and Progressions

In the TMUA, three primary terms are used when discussing ordered lists of numbers: sequence, series, and progression. A sequence is the ordered list itself. A series is the sum of the terms in a sequence. A progression is used as a neutral, catch-all term that can refer to either, helping to make question wording easier to follow.

A sequence might be defined by a formula for the $n^{th}$ term, such as $x_n = 2n + 1$ , or by a recurrence relation, which explains how to find a term based on the one before it, such as $x_{n+1} = x_n + 2$ .

Generating Terms and Spotting Patterns

The most critical skill for these problems is knowing how many terms to write out before concluding that a pattern has emerged. You must be cautious: concluding a pattern exists from only a few terms can lead to incorrect deductions. The number of terms required to verify a pattern is dictated by the structure of the recurrence relation itself.

Consider a sequence where each term depends on the previous two terms. If the rule is $x_{n+2} = f(x_n, x_{n+1})$ , you must see a repeat in two adjacent terms before you can be certain the entire sequence will begin to repeat.

Worked Example: Identifying a Recurrence Pattern

Let a sequence be generated by the recurrence relationship:
$x_0 = 1$ , $x_1 = 2$ , and $x_{n+2} = |x_n - x_{n+1}|$ for $n \geq 0$ .

Let's calculate the first several terms:

$x_0 = 1$
$x_1 = 2$
$x_2 = |x_0 - x_1| = |1 - 2| = 1$
$x_3 = |x_1 - x_2| = |2 - 1| = 1$

At this stage ( $1, 2, 1, 1$ ), it might be tempting to guess the sequence continues as $1, 2, 1, 1, 1, 1 \dots$ or repeats as $1, 2, 1, 1, 1, 2, 1, 1 \dots$ . However, because each term depends on the two preceding terms, we need more information. Let's continue:

$x_4 = |x_2 - x_3| = |1 - 1| = 0$
$x_5 = |x_3 - x_4| = |1 - 0| = 1$
$x_6 = |x_4 - x_5| = |0 - 1| = 1$
$x_7 = |x_5 - x_6| = |1 - 1| = 0$

Now the pattern is visible: $1, 2, 1, 1, 0, 1, 1, 0, 1, 1, 0 \dots$ . The sequence enters a cycle of $(1, 1, 0)$ after the first two terms.

Sigma Notation and the Fence Post Error

Sigma notation, represented by the symbol $\sum$ , is used to denote the sum of a sequence. When dealing with $\sum_{n=k}^{m} x_n$ , pay close attention to the limits. A very common mistake is the fence post error, where a student assumes the number of terms is simply $m - k$ . In reality, the number of terms is $m - k + 1$ .

Worked Example: Summing the Pattern

Using the sequence from the previous example ( $x_0 = 1, x_1 = 2, x_2 = 1, x_3 = 1, x_4 = 0 \dots$ ), find the value of $\sum_{n=0}^{100} x_n$ .

Determine the number of terms: The sum starts at $n=0$ and ends at $n=100$ . Thus, there are $100 - 0 + 1 = 101$ terms.
Identify the blocks: We write out the first few terms to see the structure: $1, 2, (1, 1, 0), (1, 1, 0), \dots$ . The first two terms ( $x_0$ and $x_1$ ) are $1 + 2 = 3$ . The remaining $101 - 2 = 99$ terms are made of repeating blocks of $(1, 1, 0)$ .
Calculate the sum: Since each block has 3 terms, there are $99 \div 3 = 33$ full blocks. The sum of each block is $1 + 1 + 0 = 2$ .
Final result: Total sum $= 3 + (33 \times 2) = 3 + 66 = 69$ .

Key takeaways

A sequence is an ordered list, while a series is the sum of that list.
For recurrence relations where $x_{n+1}$ depends on $x_n$ , a single repeating value confirms a cycle. If $x_{n+2}$ depends on $x_n$ and $x_{n+1}$ , two adjacent repeating values are required.
The number of terms in the sum $\sum_{n=k}^{m} x_n$ is always $m - k + 1$ .
Writing out the first few terms of a sigma notation expression is the best way to understand the underlying pattern.

Tips

When you encounter a complex-looking sigma notation, do not try to find a formula immediately. Write out the first four or five terms. Often, the terms will cancel each other out or form a simple repeating cycle that makes the sum easy to calculate.

Cautions

Always double check your term count for sums. In the sum from $n=1$ to $n=10$ , there are 10 terms. In the sum from $n=0$ to $n=10$ , there are 11 terms. Forgetting the '+1' is the most frequent cause of lost marks in sequence questions.

Insight

Recurrence relations can exhibit very different behaviours. Some converge to a single value, some diverge to infinity, and others enter periodic cycles. Identifying which behaviour is occurring is the first step in solving higher level progression problems.

Worked Examples

Example 1

The sequence

x_n

is given by:

x_1 = 10

x_{n+1} = \sqrt{x_n}

for

n \geq 1

. What is the value of

x_{100}

? [Note that

a^{b^{c}}

means

a^{(b^c)}

]

10^{2^{99}}

10^{2^{100}}

10^{2^{-99}}

10^{2^{-100}}

10^{-2^{99}}

10^{-2^{100}}

10^{-2^{-99}}

10^{-2^{-100}}

Practice Questions

Practice Question 1

The sequence

a_n

is given by the rule:

a_1 = 2

a_{n+1} = a_n + (-1)^n

for

n \ge 1

What is

\sum_{n=1}^{100} a_n

A:150

B:250

C:-4750

D:5150

4(1-(\frac{1}{2})^{100})

4((\frac{3}{2})^{100}-1)

Frequently asked questions

What is the difference between an explicit formula and a recurrence relation?

An explicit formula allows you to calculate the $n^{th}$ term directly by substituting the value of $n$ , such as $x_n = 3n$ . A recurrence relation defines $x_{n+1}$ in terms of $x_n$ , meaning you must usually calculate all previous terms to find a specific one.

How many terms should I write out to be sure of a pattern?

There is no fixed number, but you should write enough to see a full cycle repeat. If the rule involves the previous two terms, you must wait until a pair of consecutive terms matches a pair seen earlier.

Can a sequence have a negative index like $x_{-1}$ ?

While theoretically possible in some mathematical contexts, the TMUA generally uses natural numbers for indices, typically starting from $n=1$ or $n=0$ .

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