Sequences: patterns, nth term, recurrence relations

1️⃣ What is a Sequence?

A sequence is an ordered list of numbers. Think of it like a line of friends where each friend has a number that tells us something about them.

We write a sequence as $a_1, a_2, a_3, \dots$ where $a_n$ is the nth term (the number in the nth position).

Example: The first five terms of the sequence $1, 3, 5, 7, 9$ are $a_1=1, a_2=3, a_3=5, a_4=7, a_5=9$.

2️⃣ Spotting Patterns 🔍

Look at the differences or ratios between consecutive terms to find a pattern.

Term Number	Value	Observation
1	2	Start
2	4	+2
3	6	+2
4	8	+2

Here we see a constant difference of $+2$, so the sequence is arithmetic.

3️⃣ Arithmetic Sequences ➕

An arithmetic sequence has a constant difference $d$ between consecutive terms.

General formula: $$a_n = a_1 + (n-1)d$$

Example: For $a_1=3$ and $d=5$, the 7th term is $$a_7 = 3 + (7-1)\times5 = 3 + 30 = 33.$$

4️⃣ Geometric Sequences ➗

A geometric sequence has a constant ratio $r$ between consecutive terms.

General formula: $$a_n = a_1 r^{\,n-1}$$

Example: For $a_1=2$ and $r=3$, the 4th term is $$a_4 = 2 \times 3^{\,3} = 2 \times 27 = 54.$$

5️⃣ Recurrence Relations 🔁

A recurrence relation defines each term using one or more previous terms.

Common example: The Fibonacci sequence.

Recurrence: $$F_{n} = F_{n-1} + F_{n-2}, \quad F_1 = 1, \; F_2 = 1$$

Compute the first six terms:

1. $F_1 = 1$
2. $F_2 = 1$
3. $F_3 = F_2 + F_1 = 1 + 1 = 2$
4. $F_4 = F_3 + F_2 = 2 + 1 = 3$
5. $F_5 = F_4 + F_3 = 3 + 2 = 5$
6. $F_6 = F_5 + F_4 = 5 + 3 = 8$

Exam Tip: When you see a question about a sequence, first check if it’s arithmetic or geometric by looking at differences or ratios. If it’s neither, try to spot a recurrence relation or look for a pattern that repeats after a few terms.

6️⃣ Practice Challenge 🧩

Given the sequence: $$4, 7, 12, 19, 28, \dots$$

1. Identify the pattern.
2. Write the nth term formula.
3. Find the 10th term.

Solution:

• Differences: $3, 5, 7, 9,\dots$ – increasing by $2$ each time.
• Recurrence: $a_{n} = a_{n-1} + (2n-1)$ with $a_1 = 4$.
• 10th term: $a_{10} = 4 + \sum_{k=2}^{10}(2k-1) = 4 + 81 = 85$.

Exam Tip: For sequences that don’t fit the standard arithmetic or geometric form, look for a pattern in the differences or ratios. If the differences themselves form a simple sequence (like consecutive odd numbers), that often hints at a recurrence relation.