Grade 12 Mathematics · Unit 1 · Chapter 1.1 · QuickNotes

Sequences

These QuickNotes give you the most important points to remember for the ESSLCE. They are based on the MoE Grade 12 Mathematics Textbook, Unit 1, Chapter 1.1.

~6 min read
Summary
  • A sequence is a list of numbers written in a fixed order, where each number follows a clear rule.
  • The rule is often given as a general term an, so you find any term by putting its position number n into the rule.
  • A recursion formula builds each new term from the terms before it, and the Fibonacci and Mulatu sequences work this way.

Key Words

  • Sequence: a sequence is an ordered list of numbers that follows a rule.
  • Term: a term is one single number in the sequence, and a1 is the first term, a2 is the second term, and so on.
  • General term (nth term): the general term an is the rule that gives any term once you know its position n.
  • Finite sequence: a finite sequence is one that has a last term, so it stops.
  • Infinite sequence: an infinite sequence is one that has no last term, so it goes on forever.
  • Recursion formula: a recursion formula gives each term by using one or more terms that came before it.

What This Chapter Is About

  • This chapter teaches you what a sequence is and how to find its terms from a rule.
  • You also learn to draw the graph of a sequence and to recognise two famous sequences that are built by recursion.

What a Sequence Is

  • A sequence is a function whose inputs are whole-number positions, usually starting at 1, and whose outputs are the terms.
  • This means each position number n is matched to exactly one term an, just like a machine that turns 1, 2, 3 into the first, second, and third terms.
  • We write the terms as a1, a2, a3, and the general term an is the rule for the term in position n.
  • The order matters, so 1, 3, 5 is a different sequence from 5, 3, 1.

Finite and Infinite Sequences

  • A finite sequence has a last term, so its positions are 1, 2, 3, up to some final n.
  • An infinite sequence has no last term, so its positions are all the natural numbers and the three dots show it never ends.
  • For example, 2, 4, 6, 8 is finite, but 2, 4, 6, 8, … is infinite.

Finding Terms From the Rule

  • When you are given the general term an, you find any term by putting its position number into the rule.
  • To list the first few terms, you simply use n = 1, then n = 2, then n = 3, and continue.

Worked example: list the first three terms of an = 2n − 1.

Put n = 1, 2, 3 into the rule: a1 = 2(1) − 1 = 1, a2 = 2(2) − 1 = 3, and a3 = 2(3) − 1 = 5. So the first three terms are 1, 3, 5.

The Graph of a Sequence

  • To draw a sequence, you plot each pair (n, an), with the position n along the bottom and the term value an going up.
  • The graph is a set of separate dots, not a solid line, because a sequence only has values at whole-number positions.
  • The dots still follow the shape of the rule, so a rule like an = 2n + 1 gives dots that lie along a straight-line pattern.
1 2 3 4 5 6 7 n (term number) 3 5 7 9 11 13 15 a n a   =   2 n   +   1 n

Figure 1.1: the graph of a sequence is a set of separate dots, and here they follow a straight-line pattern.

Fibonacci and Mulatu Sequences

The Fibonacci sequence

  • The Fibonacci sequence starts with 1, 1 and then each new term is the sum of the two terms before it.
  • This gives 1, 1, 2, 3, 5, 8, 13, 21, and so on, because 1 + 1 = 2, then 1 + 2 = 3, then 2 + 3 = 5.
  • It is named after the Italian mathematician Leonardo Fibonacci, and the same pattern appears in the growth of leaves and the breeding of rabbits.

The Mulatu sequence

  • The Mulatu sequence was introduced by Professor Mulatu Lemma, an Ethiopian mathematician, so it is a proud part of our own mathematics.
  • It starts with 4, 1 and then, like Fibonacci, each new term is the sum of the two terms before it.
  • This gives 4, 1, 5, 6, 11, 17, 28, and so on, because 4 + 1 = 5, then 1 + 5 = 6, then 5 + 6 = 11.

Recursion Formulas

  • A recursion formula defines a term by using the terms that come before it, so you must know earlier terms to find the next one.
  • The Fibonacci and Mulatu sequences are both recursive, because each term needs the two terms before it.
  • A general-term rule like an = 2n + 1 is different, because it gives any term directly from its position without needing the earlier terms.

Quick Comparison Table

Feature General-term rule Recursion formula
How it gives a term Directly from the position n From one or more earlier terms
Need earlier terms first? No Yes
Example an = 2n + 1 Fibonacci and Mulatu

Common Mistakes to Avoid

  • Do not confuse the position number n with the term value an, because n is where the term sits and an is the term itself.
  • Do not join the dots of a sequence into a solid line, since a sequence only has values at whole-number positions.
  • Do not use a recursion formula without knowing the starting terms first, because each new term needs the earlier ones.
  • Always read whether the count starts at n = 0 or n = 1, because this changes which term is first.

Easy Ways to Remember

  • A sequence is a machine: you feed in a position number n and it gives back one term an.
  • For Fibonacci and Mulatu, just remember “add the two before” to get the next term.
  • Think of recursion as a chain, because each link is built from the links right before it.

Quiz

Tap an answer to check it.

1. What is the third term of the sequence an = 3n − 1?

2. Which sequence is finite?

3. In the Fibonacci sequence 1, 1, 2, 3, 5, 8, …, what is the next term?

4. The graph of a sequence is best shown as which of these?

5. Which sequence is defined by recursion?

Remember: A sequence is an ordered list of numbers that follows a rule. Use the general term an to find any term directly from its position, and use a recursion formula, like the Fibonacci and Mulatu sequences, when each term is built from the terms before it.

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