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Unit B1 Section 3
Estimating Number Patterns

A formula or rule for extending a sequence can be used to work out any term of a sequence without working out all the terms. For example, the 100th term of the sequence,

1, 4, 7, 10, 13, ...

can be calculated as 298 without working out any other terms.

Worked Examples

1

Find the 20th term of the sequence

8, 16, 24, 32, ...

The terms of the sequence can be obtained as shown below.

1st term = 1 × 8 = 8
2nd term = 2 × 8 = 16
3rd term = 3 × 8 = 24
4th term = 4 × 8 = 32

This pattern can be extended to give

20th term = 20 × 8 = 160
2

Find the 10th and 100th terms of the sequence

3, 5, 7, 9, 11, ...

The terms above are given by

1st term = 3
2nd term = 3 + 2 = 5
3rd term = 3 + 2 × 2 = 7
4th term = 3 + 3 × 2 = 9
5th term = 3 + 4 × 2 = 11

This can be extended to give

10th term = 3 + 9 × 2 = 21
100th term = 3 + 99 × 2 = 201
3

Find the 20th term of the sequence

2, 5, 10, 17, 26, 37, ...

The terms of this sequence can be expressed as

1st term = 12 + 1
2nd term = 22 + 1
3rd term = 32 + 1
4th term = 42 + 1
5th term = 52 + 1

Extending the pattern gives

20th term = 202 + 1 = 401

Exercises

Find the 10th and 20th terms of each sequence below.

(a)
4, 8, 12, 16, 20, ..., , ...,
(b)
5, 10, 15, 20, 25, ..., , ...,
(c)
11, 21, 31, 41, 51, ..., , ...,
(d)
7, 9, 11, 13, 15, ..., , ...,
(e)
5, 9, 13, 17, 21, ..., , ...,
(f)
20, 19, 18, 17, 16, ..., , ...,
(g)
50, 44, 38, 32, 26, ..., , ...,
(h)
22, 25, 28, 31, ..., , ...,
(i)
8, 7, 6, 5, 4, ..., , ...,
(j)
−4, 0, 4, 8, 12, ..., , ...,
(k)
7, 12, 17, 22, 27, ..., , ...,
(l)
3, −2, −7, −12, ..., , ...,

(a)

Find the 10th term for each of the two sequences below.

(i)
3, 6, 11, 18, 27, ...
(ii)
5, 6, 7, 8, 9, ...
(b)

Hence find the 10th term of the sequences,

(i)
8, 12, 18, 26, 36, ...
(ii)
15, 36, 77, 144, 243, ...
(iii)
−2, 0, 4, 10, 18, ...

For each sequence of shapes below find the number of dots in the 10th shape.

(a)
dots
(b)
dots
(c)
dots

Patterns of triangles are made using sticks. The first three patterns are drawn below.

(a)

How many sticks has Pattern 4?

sticks
(b)

A pattern needs 233 sticks. What is the number of this pattern?

Pattern
(c)
(i)

How many sticks are needed to make Pattern 100?

sticks
(ii)

Explain how you found your answer.

nth pattern needs sticks