 1. Arithmetic sequence
3. Cubic sequence
4. Geometric sequence

An arithmetic or linear sequence is an ordered list of numbers where the difference between two consecutive terms is constant. To find missing terms we can adjust known ones by the common difference or a multiple of it. A quadratic sequence is an ordered list of numbers in which the second difference between terms is constant. To find the term to term rule, identify the common second difference between consecutive terms and use this to evaluate the first differences. A cubic sequence is an ordered list of numbers in which the third difference between terms is constant. To find the term to term rule, identify the third difference between consecutive terms and use this to evaluate the second and first differences. In a geometric sequence, all terms are multiplied or divided by the same value. This value is called the common ratio and can be used to find the next term in a sequence. The common ratio is determined by dividing one term by the previous term. # 1

Julie is building houses of cards. She notices the number of cards she needs to build houses of different heights follows a quadratic sequence. Given a one storey house of cards needs 3 cards, a two storey needs 8 cards, and three storeys needs 15 cards, find the number of cards Julie needs for a house of cards 4 storeys high.

Identify n₁ = 3, n₂ = 8 and n₃ = 15.
The sequence of first differences is given by {5, 7}.
The common second difference is 2.
The first difference between the 3rd term and the 4th term is 7 + 2 = 9.
n₅ = 15 + 9 = 24

24 # 2

Louise is part of her school’s football team. She notices that with each training session, the number of keepie uppies she can do increases by 7. Given she could do 2 keepie uppies in her first training session, how many will she be able to do on her twelfth training session?

The number of keepie uppies Louise can do is an arithmetic sequence, since there is a constant difference in how many she can do between each consecutive training session.
The common difference is given to be d = 7.
We need to add 11 differences to the first term to get the 12th term.
a₁₂ = 2 + 11 × 7 = 79

79 # 3

Explain why the sequence {1, 5, 9, 13, 17, …} is not cubic.

Finding the first difference between terms gives a constant difference of 4, hence this is a linear sequence.

constant first difference # 4

Nathan is playing space invaders on his console. He realizes that the number of points he scores with each game follows a quadratic sequence. Given Nathan scored 66 on his second game, and 147 on his third and 260 on the fourth, calculate how many points he has got in his first game.

Identify that n₂ = 66, n₃ = 147 and n₄ = 260.
The sequence of first differences is given by {81, 113}.
The common second difference is 32.
The first difference between the 1st term and the 2nd term is 81 − 32 = 49.
n₁ = 66 − 49 = 17

17 # 5

Determine the next term of the geometric sequence {8, −16, 32, −64, …}.

The common ratio is given by r = −16 ÷ 8 = −2.
a₅ = −64 × (−2) = 128

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