6. Number Patterns and Recursions

Worksheet

1

Write down the next two terms for each arithmetic sequence:

a

4, 8, 12, 16

b

2, 3.5, 5, 6.5

c

6, 2, - 2, - 6

d

- 8, - \dfrac{23}{3}, - \dfrac{22}{3}, - 7

2

State the common difference for each arithmetic sequence:

a

- 6, - \dfrac{39}{7}, - \dfrac{36}{7}, - \dfrac{33}{7}\ldots

b

330, 280, 230, 180\ldots

3

For each of the following arithmetic sequences:

i

Write the first four terms of the sequence.

ii

Find the common difference.

a

T_n = 3 n + 8

b

T_n = 11 + \left(n - 1\right) \times 10

c

T_n = - 7 - 3 \left(n - 1\right)

4

Write the first four terms in each of the following arithmetic sequences:

a

The first term is - 10 and the common difference is 4.

b

The first term is - 8 and the common difference is - 2.

c

The first term is a and the common difference is d.

5

Find the missing terms in the following arithmetic sequences:

a

8, ⬚, 16, 20, ⬚

b

⬚, 0, ⬚, 10, ⬚

6

a

Determine whether each set of numbers is an arithmetic sequence:

i

1, \sqrt{5}, 5, 5 \sqrt{5}, \ldots

ii

2, 2^{2}, 2^{4}, 2^{6}, \ldots

iii

2, 0, - 2, - 4, \ldots

iv

3, - 3, 3, - 3, \ldots

v

3, 6, 12, 24\ldots

vi

5, 7, 5, 7\ldots

b

State the common difference of the arithmetic sequence found in part (a).

7

For each general formula of an arithmetic sequence:

i

Determine a, the first term in the arithmetic sequence.

ii

Determine d, the common difference.

iii

Find the indicated term in the sequence.

a

T_n = 15 + 5 \left(n - 1\right);\enspace T_9

b

T_n = - 8 n + 28;\enspace T_5

8

Consider the first three terms of the following arithmetic sequences:

i

Find the common difference, d.

ii

State T_n, the general rule for the nth term in the sequence.

iii

Find the indicated term in the sequence.

a

5, 12, 19, \ldots ;\enspace T_{10}

b

17, 16.2, 15.4, \ldots ;\enspace T_{13}

c

10, 3, - 4, \ldots ; \enspace T_{9}

d

5, \dfrac{23}{4}, \dfrac{13}{2}, \ldots ;\enspace T_{14}

9

Consider the arithmetic sequence: 1.4, 2.3, 3.2, \ldots, 10.4

a

Determine d, the common difference.

b

Solve for n, the number of terms in the sequence.

10

Find the missing 5 terms in the arithmetic sequence which has - 12 as its first term and 24 as its last term:

- 12,\, ⬚,\, ⬚,\, ⬚,\, ⬚,\, ⬚,\, 24

11

Find the value of x such that x + 4, 6 x + 5, and 9 x - 8 form successive terms in an arithmetic sequence.

12

In an arithmetic sequence, T_5 = 21 and T_{19} = 77.

a

Find the value of d.

b

Find the value of a.

c

State the general rule for T_n.

d

Find T_{11}.

13

In an arithmetic sequence, T_7 = 44 and T_{14} = 86.

a

Find the value of d.

b

Find the value of a.

c

State the general rule for T_n.

d

Hence, find T_{25}.

14

In an arithmetic sequence, T_7 = 9 and T_{15} = 13.

a

Find the value of d.

b

Find the value of a.

c

State the general rule for T_n.

d

Find T_{26}.

15

In an arithmetic sequence with common difference d, the first term is 32.

a

Write a simplified expression for the 5th term.

b

Write a simplified expression for the 9th term.

c

Given that the 9th term is 4 times the 5th term, find the common difference d.

16

The first three terms of an arithmetic sequence are: 82, 75, 68, \ldots

a

Determine the number of positive terms in the sequence.

b

Find the last positive term in the sequence.

17

For each of the following sequences, find the value of n:

a

0.9, 1.5, 2.1, \ldots where T_n = 22.5

b

2, 7, 12, \ldots where T_n = 132

c

2, - 3 , - 8 , \ldots where T_n = - 578

d

5, \dfrac{17}{4}, \dfrac{7}{2}, \ldots where T_n = - 37

18

The nth term of an arithmetic sequence is T_n = - 530.

Find the value of n given that T_1 = 28 and d = - 18.

19

In an arithmetic sequence the 6th term is x and the 10th term is y.

a

Form an expression for d in terms of x and y.

b

Form an expression for a in terms of x and y.

c

Form an expression for the 16th term in terms of x and y.

20

Consider the recurrence relation: u_{n + 1} = u_n + 3 with initial term u_1.

a

Complete the table identifying the first 4 terms of the sequence in terms of u_1:

b

Express u_7 in terms of u_1.

c

Express u_n in terms of u_1.

n | 1 | 2 | 3 | 4 |
---|---|---|---|---|

u_n |

21

The nth term of an arithmetic sequence is given by the equation T_n = 12 + 4 \left(n - 1\right).

a

Complete the table of values:

b

By how much are the consecutive terms in the sequence increasing?

n | 1 | 2 | 3 | 4 | 10 |
---|---|---|---|---|---|

T_n |

c

Plot the points in the table on a coordinate plane.

d

If the points on the graph were joined, would they form a straight line or a curve?

22

The nth term of an arithmetic sequence is given by the equation T_n = 15 - 5 \left(n - 1\right).

a

Complete the table of values:

b

Find the difference between consecutive terms.

n | 1 | 2 | 3 | 4 | 10 |
---|---|---|---|---|---|

T_n |

c

Plot the points in the table on a coordinate plane.

d

If the points on the graph were joined, would they form a straight line or a curve?

23

Each given table of values represents terms in arithmetic sequence. For each table:

i

Find d, the common difference.

ii

Write a simplified expression for the general term, T_n.

iii

Find the missing term in the table.

a

n | 1 | 2 | 3 | 4 | 15 |
---|---|---|---|---|---|

T_n | 9 | 15 | 21 | 27 |

b

n | 1 | 2 | 3 | 4 | 10 |
---|---|---|---|---|---|

T_n | 5 | -4 | -13 | -22 |

c

n | 1 | 2 | 3 | 4 | 11 |
---|---|---|---|---|---|

T_n | 4 | \dfrac{17}{2} | 13 | \dfrac{35}{2} |

d

n | 1 | 3 | 5 | 7 | 19 |
---|---|---|---|---|---|

T_n | -10 | -26 | -42 | -58 |

e

n | 1 | 6 | 10 | 18 |
---|---|---|---|---|

T_n | -10 | -40 | -64 |

24

The values in the table show terms in an arithmetic sequence for values of n.

Complete the table.

n | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

T_n | -6 | -26 |

25

The plotted points represent terms in an arithmetic sequence.

a

Complete the table of values for the given points:

n | 1 | 2 | 3 | 4 |
---|---|---|---|---|

T_n |

b

State the value of d, the common difference.

c

Write a simplified expression for the general term, T_n.

d

Find the 14th term of the sequence.

26

The plotted points represent terms in an arithmetic sequence.

a

Complete the table of values for the given points:

n | 1 | 3 | 5 |
---|---|---|---|

T_n |

b

State the value of d, the common difference.

c

Write a simplified expression for the general term, T_n.

d

Find the 18th term of the sequence.

27

The plotted points represent terms in an arithmetic sequence.

a

State the value of d, the common difference.

b

Write a simplified expression for the general term, T_n.

c

Find the gradient of the line that passes through these points.

28

The plotted points represent terms in an arithmetic sequence.

a

State the value of d, the common difference.

b

Write a simplified expression for the general term, T_n.

c

The points are reflected about the horizontal axis to form three new points.

If these new points represent consecutive terms of an arithmetic sequence, write the equation for T_k, the kth term in this new sequence.

29

A diving vessel descends below the surface of the water at a constant rate so that the depth of the vessel after 1 minute, 2 minutes and 3 minutes is 50 metres, 100 metres and 150 metres respectively.

a

By how much is the depth increasing each minute?

b

Find the depth of the vessel after 4 minutes.

c

Write a recursive rule to define the vessel's depth as an arithmetic sequence in terms of d_{n+1} with first term d_1.

d

Find the depth of the vessel after 10 minutes.

30

For a fibre-optic cable service, Christa pays a one off amount of \$200 for installation costs and then a monthly fee of \$30.

a

Complete the table of values for the total cost \left(T\right) of Christa's service over n months.

b

By how much are consecutive terms in the sequence increasing?

n | 1 | 2 | 3 | 4 | 18 |
---|---|---|---|---|---|

T |

c

Considering the table of values, plot the points corresponding to n = 1, 2, 3 and 4.

d

If the points on the graph were joined, would they form a straight or curved line?

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Define and explain key concepts in use of a first-order linear recurrence relation to generate the terms of a number sequence, and apply a range of related mathematical routines and procedures

Define and explain key concepts in generation of an arithmetic sequence using a recurrence relation, tabular and graphical display; and the rule for the nth term of an arithmetic sequence and its evaluation, and apply a range of related mathematical routines and procedures