Can you think of any other kinds of sequences? List out five different types of sequences and discuss their properties with your friends.

Step-by-step solution

Idea: Sequences can be sorted by how each term relates to the previous one (add, subtract, multiply, combine earlier terms), by whether they increase or decrease, and by whether they stop (finite) or go on (infinite).

1. Increasing by a fixed amount: 5, 10, 15, 20, … (multiples of 5). Property: the difference between neighbours is always 5; the terms grow steadily.½ mark
2. Decreasing: 100, 90, 80, 70, … or 1, 12, 13, 14, … Property: each term is smaller than the one before; the terms can become negative or get closer and closer to 0.½ mark
3. Multiplying by a fixed number: 2, 4, 8, 16, 32, … Property: each term is double the previous one, so the terms grow very fast.½ mark
4. Alternating signs: 1, −1, 1, −1, … or 2, −4, 8, −16, … Property: the terms switch between positive and negative.½ mark
5. Repeating (periodic): 1, 2, 3, 1, 2, 3, … like the days of the week. Property: the same block of terms repeats; it neither grows nor shrinks.½ mark
6. Other kinds to discuss: a constant sequence 7, 7, 7, …; sequences where each term uses the two previous terms, 1, 1, 2, 3, 5, 8, …; and sequences with no simple rule, such as the primes 2, 3, 5, 7, 11, ….½ mark

Answer to write in the exam

Common mistakes that cost marks

• Listing five examples of the same kind (e.g. 2, 4, 6 …; 3, 6, 9 …; 5, 10, 15 …). These are all ‘add a fixed number’; choose sequences that behave differently.
• Giving a list without a rule or order, such as ‘my favourite numbers’. A sequence must be in a definite order, and its properties should be clear.
• Saying a sequence ‘has no pattern’ when the terms simply decrease or alternate. Decreasing and alternating sequences still follow a rule.

How this can come in the exam

Try one yourself

Write one example each of a constant sequence and an alternating sequence, and give the next two terms of each.

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• Consider the expression u_n = 2n − 1. This states that the nth term of the sequence is given by the rule 2n − 1.
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• Using the explicit rule u_n = 2n − 1, find the 53rd term, the 108th term, and the 1170th term of the odd number sequence.
• Consider the sequence that is generated by the explicit formula s_n = 5n − 2.
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