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Middle Years

2.08 Number patterns and sequences

Worksheet
Sequences
1

Determine whether the following patterns are increasing, decreasing or neither:

a

7, 6, 11, 3, 9

b

25, 22, 19, 16, 13

c

6.5, 6.6, 6.7, 6.8, 6.9

d

5, 4, 9, 1, 7

e

21, 20, 19, 18, 17

f

5.4, 9.4, 6.2, 6.6, 7.0

2

For each of the following patterns:

i

State whether the pattern is increasing or decreasing.

ii

Determine by how much the pattern is increasing or decreasing each time.

a

9.5, 9.7, 9.9, 10.1, 10.3

b

15.3, 14.7, 14.1, 13.5, 12.9

c

31, 25, 19, 13

d

15, 21, 27, 33

e

5, 12, 19, 26

f

34, 28, 22, 16

g

17, 23, 29, 35

h

9, 16, 23, 30

3

Find the next number in the following sequences:

a

2,4, 6, 8

b

7, 9, 11, 13

c

65, 58, 51, 44

d

16, 36, 64, 100, 144

e

2,\, 2\dfrac{1}{2},\, 3,\, 3\dfrac{1}{2}

f

1,\, 1\dfrac{1}{3},\, 1\dfrac{2}{3},\, 2

4

State the third term in the sequence: 4, -5, 6, -7, 8, \ldots

5

Consider the sequence a_n. If the first term is a_1, write the notation used for the fifth term of the sequence.

6

Determine whether each sequence is finite or infinite:

a

1, 2, 3, 4, 5, 6

b

-1, -2, -3, -4, -5, -6\text{, . . .}

c

-1, -2, -3, -4, -5

d

1, 2, 3, 4, 5, 6\text{, . . .}

7

If all the dates in July were used to form a sequence, would the sequence be finite or infinite?

8

If all the years from 1966 onwards were used to form a sequence, would the sequence be finite or infinite?

Explicit rules
9

State the first five terms of the following sequences. Round terms to four decimal places where necessary:

a

a_n = 3 n - 3

b

a_n = - 2^{n}

c

a_n = 2^{n + 1}

d

a_n = \left( - 2 \right)^{n} \left( 3 n\right)

e

a_n = \dfrac{1}{\sqrt{n}}

f

a_n = \left(1 + \dfrac{1}{n}\right)^{n}

10

State the first four terms, in exact form, of the following sequences:

a

a_n = \dfrac{n + 4}{n + 3}

b

a_n = \dfrac{5 n - 1}{n^{2} + 5}

c

a_n = \left(\dfrac{1}{2}\right)^{n} \left(n - 3\right)

d

a_n = \left( - 1 \right)^{n - 3} \left(n - 1\right)

11

Consider the rule: "The starting number, N, is doubled, then 4 is subtracted to get the answer A."

a

Use the rule to complete the table of values:

\text{Starting Number }(N)12131415
\text{Answer }(A)
b

Write the rule for A in terms of N.

12

Consider the rule: "The starting number, N, has 9 added to it. The sum is then multiplied by 5 to get the answer A."

a

Use the rule to complete the table of values:

\text{Starting Number }(N)4567
\text{Answer }(A)
b

Write the rule for A in terms of N.

Patterns
13

Each of the patterns below was created in steps using matchsticks.

a

Complete the following table of values for each of the given patterns:

\text{Step number} \left(t\right)1234510
\text{Number of matchsticks}\left(m\right)
b

Write a formula that describes the relationship between the number of matches, m, and the step number, t.

a
b
c
d
14

Consider the following pattern:

a

If s is the total number of squares for figure number n, write an algebraic rule that gives s in terms of n.

b

Find the number of squares when n = 9.

15

Vanessa is making a sequence of shapes out of tiles. She creates a table comparing the sequence number of a shape to the number of tiles needed to make it:

\text{Sequence} \\ \text{number } \left(n\right)123456
\text{Number of} \\ \text{tiles } \left(T\right)357
a

How many new tiles are added at each step?

b

Find how many tiles Vanessa will need to make the next three shapes in the sequence by completing the table of values.

c

An equation to represent the relationship between a shape's sequence number and the number of tiles needed can be written in the form T=s+(n-1)d, where s is the starting number of tiles and d is the number of new tiles added each step.

i
Write an equation to represent the relationship between a shape's sequence number and the number of tiles needed in the form given.
ii
Rewrite the equation found in the form T=an+b.
d

Find how many tiles Vanessa will need to make the 20th shape in the sequence.

16

Consider the pattern of images below:

a

If t represents the number of shaded triangles, write an expression for t in terms of n.

b

Find the value of t when n = 8.

17

Christa deposits \$1 into a new bank account while Jenny deposits \$38 into a new bank account. The next day, Christa adds \$2 to the account, and doubles how much she adds each day thereafter. Jenny adds \$5 each day.

a

List the amount Christa had in her account each day for the first four days.

b

List the amount Jenny had in her account each day for the first four days.

c

On which day will they have the same amount of money in their bank accounts?

18

Triangular numbers count the number of objects that make up an equilateral triangle. The first three triangular numbers are shown below:

a

Complete the table, showing the first 6 triangular numbers:

n123456
S136
b

The number of spots needed to represent the nth triangular number is given by the formula S = \dfrac{n \left(n + 1\right)}{2}. Find the number of spots needed to represent the 10th triangular number.

19

A population of rabbits are counted in an area, and it is found that the population is doubling each month. The population counts at the end of each of the first three months are shown below:

a

Complete the table, showing the population at the end of each of the first 5 months:

n12345
P
b

If the population continues to grow in this manner, the population at the end of the nth month would be given by P = 2^{n}. Find the population at the end of 1 year.

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