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3.04 Arithmetic progressions - calculator assumed

Worksheet
Arithmetic sequences
1

Use the following recursive definitions to state the first 3 terms of the sequence:

a

T_{n + 1} = T_n + 5, T_{1} = 4

b

T_n = T_{n - 1} - 4, T_{1} = - 12

c

T_{n + 1} = - 2.5 + T_n, T_{1} = 4

d

T_n = 13 + T_{n - 1}, T_{1} = 3

2

For the following arithmetic progressions given by the explicit rule:

i

Find a, the first term in the arithmetic progression.

ii

Find d, the common difference.

iii

Calculate T_9.

a

T_n = 4 + 5 \left(n - 1\right)

b

T_n = 2 - 6 \left(n - 1\right)

c

T_n = - 2 + 6 \left(n - 1\right)

d

T_n = - 4 - 5 \left(n - 1\right)

3

For each of the following sequences:

i

State the recursive rule for T_n in terms of T_{n - 1}, as well as the initial term T_1.

ii

State the explicit rule for T_n in terms of n.

iii

Calculate T_{16}.

a

12, 15, 18, 21, \ldots

b

22, 17, 12, 7, \ldots

c

- 20 , - 16 , - 12 , - 8 , \ldots

d

5, 6.5, 8, 9.5, \ldots

4

For each of the sequences below:

i

Find d, the common difference.

ii

Write a recursive rule for T_n in terms of T_{n - 1} and an initial condition for T_1.

a

The first term of an arithmetic sequence is 5. The third term is 17.

b

The first term of an arithmetic sequence is 6. The fourth term is - 6.

5

Find the value of n if:

a

The nth term of the sequence 23, 14, 5, - 4 , \ldots is -238.

b

The nth term of the sequence \dfrac{2}{3}, \dfrac{11}{12}, \dfrac{7}{6}, \dfrac{17}{12}, \ldots is \dfrac{49}{6}.

6

In an arithmetic progression where a is the first term, and d is the common difference, we have T_2 = 9 and T_5 = 27.

a

Find d, the common difference.

b

Find a, the first term in the sequence.

c

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

d

Hence, calculate T_{30}.

7

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

Arithmetic sequences in tables and graphs
8

For each of the following graphs of sequences, state the recursive rule for T_n in terms of T_{n-1} as well as the initial term T_1:

a
1
2
3
4
5
6
n
2
4
6
8
10
12
14
16
18
20
T_n
b
1
2
3
4
5
6
n
2
4
6
8
10
12
14
16
18
20
T_n
c
1
2
3
4
5
6
n
-20
-18
-16
-14
-12
-10
-8
-6
-4
-2
2
4
6
8
10
T_n
d
1
2
3
4
5
6
n
1
2
3
4
5
6
7
8
9
10
T_n
9

For the following sequences given by the recursive rule:

i

State the explicit rule for T_n in terms of n.

ii

Complete a table of values of the form:

n123410
T_n
iii

Plot the points from the table of values on a graph.

iv

Determine whether each point on the graph in part (c) would form a straight line.

a

T_{n + 1} = T_n + 3, T_1 = 5

b

T_{n + 1} = T_n + 4, T_1 = - 46

c

T_{n + 1} = T_n + 2.5, T_1 = 5

d

T_{n + 1} = T_n - 4, T_1 = 50

10

For the following graphs of arithmetic sequences:

i

Complete the table of values.

n1234
T_n
ii

Find the value of d.

iii

Write the general rule for the nth term of the sequence, T_n.

iv

Find the 12th term of the sequence.

a
1
2
3
4
n
1
2
3
4
5
6
7
8
9
T_n
b
1
2
3
4
5
n
-13
-12
-11
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
T_n
11

For the following arithmetic sequences represented by a table of values:

i

Find d, the common difference.

ii

Write the general rule for the nth term of the sequence, T_n.

iii

Find T_{21}.

a
n1234
T_n5111723
b
n1234
T_n62-2-6
12

For the sequence given by the recursive rule T_{n + 1} = T_n - 4, T_1 = 2:

a

Plot the first six points of the sequence on a graph.

b

State the explicit rule for T_n in terms of n.

c

State the first position n where the sequence becomes less than or equal to - 400.

13

For the sequence given by the recursive rule T_{n + 1} = T_n + 4, T_1 = - 1:

a

Plot the first six points of the sequence on a graph.

b

State the explicit rule for T_n in terms of n.

c

State the first position n where the sequence becomes greater than or equal to 500.

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Outcomes

3.2.1

use recursion to generate an arithmetic sequence

3.2.2

display the terms of an arithmetic sequence in both tabular and graphical form and demonstrate that arithmetic sequences can be used to model linear growth and decay in discrete situations

3.2.3

deduce a rule for the nth term of a particular arithmetic sequence from the pattern of the terms in an arithmetic sequence, and use this rule to make predictions

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