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5.01 Arithmetic sequences

Worksheet
Arithmetic sequences
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

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

a

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

b

The first term is 3 and the common difference is - 5.

c

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

4

Find the missing terms in the following arithmetic sequences:

a

5 , ⬚ , -1 , - 4 , ⬚

b

⬚, - 11, ⬚, - 19, ⬚

5
a

Which of the following sets of numbers is an arithmetic sequence:

A
2, - 2, 2, - 2 , \ldots
B
5, 5^{2}, 5^{4}, 5^{6} , \ldots
C
1, \sqrt{2}, 2, 2 \sqrt{2} , \ldots
D
5, 3, 1, -1 , \ldots
E
3, 6, 12, 24 , \ldots
F
5, 7, 5, 7 , \ldots
b

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

6

The first term of an arithmetic progression is 3 and the common difference is 4.

a

Find the 15th term of the sequence.

b

Is 202 a term of the sequence?

c

Find the first term of the sequence that exceeds 1000.

Recurrence relation and explicit formula
7

The nth term in an arithmetic sequence is given by the formula T_{n} = 14 + 5 \left(n - 1\right).

a

Find a, the first term in the sequence.

b

Find d, the common difference.

c

Find the 10th term in the sequence.

8

The nth term in an arithmetic sequence is given by the formula T_{n} = - 9 n + 19.

a

Find a, the first term in the sequence.

b

Find d, the common difference.

c

Find the 9th term in the sequence.

9

For each of the following general terms of an arithmetic sequence:

i

Write down the first four terms.

ii

Find the common difference.

a

T_{n} = 11 + \left(n - 1\right) \times 10

b

T_{n} = - 7 - 3 \left(n - 1\right)

c

T_{n} = 3 n + 8

10

For each of the following arithmetic sequences:

i

Find d, the common difference.

ii

State the equation for finding T_{n}, the nth term in the sequence.

iii

Find the 9th term in the sequence.

a

6, 16, 26 , \ldots

b

8, -1, - 10 , \ldots

11

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

- 12,\, ⬚,\, ⬚,\, ⬚,\, ⬚,\, ⬚,\, 24

12

Find the value of x such that x + 2, 6 x + 5, and 9 x + 14 form successive terms in an arithmetic sequence.

13

Consider the arithmetic sequence: 1.7, \, 2.5, \, 3.3, \, \ldots , \, 12.9

a

Find d, the common difference.

b

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

14

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

15

The nth term of an arithmetic sequence is T_{n} = - 493.

Find the value of n given that T_{1} = 24 and d = - 11.

16

The first three terms of an arithmetic sequence are: 46, 39, 32, \ldots

a

Find the range of values of n for which the terms in the sequence are positive.

b

Hence, determine the number of positive terms in the sequence.

c

Find the last positive term in the sequence.

17

In an arithmetic sequence, T_{5} = 13 and T_{19} = 41.

a

Find the value of d.

b

Find the value of a.

c

Find the 13th term in the sequence.

18

In an arithmetic sequence, T_{4} = 10.1 and T_{13} = 16.4.

a

Find the value of d.

b

Find the value of a.

c

Find the 29th term in the sequence.

19

In an arithmetic sequence, T_{7} = 16 and T_{10} = 22.

a

Find d, the common difference.

b

Find a, the first term in the sequence.

c

State the equation for T_{n}.

d

Find the 30th term in the sequence.

20

In an arithmetic sequence, the first term is 3.

a

Find an expression for the 3rd term.

b

Find an expression for the 10th term.

c

Given that the 10th term is 4 times the 3rd term, find the common difference d.

21

In an arithmetic sequence, the 6th term is x and the 14th 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

Hence, find an expression for the 20th term in terms of x and y.

22

The first term of an arithmetic sequence is 2. The fifth term is 26.

a

Find d, the common difference.

b

Write a recursive rule for T_{n} in terms of T_{n - 1} which defines this sequence and an initial condition for T_{1}.

23

Consider the arithmetic sequence with terms T_{3} = 14 and T_{12} = 59.

a

Find d, the common difference.

b

Find the term T_{1}.

c

Write a recursive rule for T_{n} in terms of T_{n - 1} which defines this sequence and an initial condition for T_{1}.

Tables and graphs
24

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

Calculate the difference between consecutive terms.

n123412
T_n
c

Plot the points in the table on a number plane.

d

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

25

The nth term of an arithmetic sequence is given by the equation T_{n} = 10 - 10 \left(n - 1\right).

a

Complete the table of values.

b

Calculate the difference between consecutive terms.

n123414
T_n
c

Plot the points in the table on a number plane.

d

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

26

Each of the following 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 12th term of the sequence.

a
n1234
T_n5121926
b
n1234
T_n3\dfrac{24}{5}\dfrac{33}{5}\dfrac{42}{5}
27

The plotted points represent terms in an arithmetic sequence:

a

Complete the table of values for the given points:

n1234
T_{n}
b

Find d, the common difference.

c

Write a simplified expression for the general term, T_{n}.

d

Find the 15th term of the sequence.

1
2
3
4
n
2
4
6
8
10
12
14
16
18
20
T_{n}
28

The plotted points represent terms in an arithmetic sequence:

a

Complete the table of values for the given points:

n159
T_n
b

Find d, the common difference.

c

Write a simplified expression for the general term, T_{n}.

d

Find the 20th term of the sequence.

1
2
3
4
5
6
7
8
9
n
-4
-2
2
4
6
8
10
12
14
16
18
20
22
T_{n}
29

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

Complete the table.

n12345
T_n-6-26
30

The plotted points represent terms in an arithmetic sequence:

a

Find d, the common difference.

b

Write a simplified expression for the general term, T_{n}.

c

The points are reflected across 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.

1
2
3
4
n
1
2
3
4
5
6
7
8
9
10
11
12
T_{n}
Applications
31

When a new school first opened, a students started at the school. Each year, the number of students increases by the same amount, d.

a

At the beginning of its 5th year, it had 393 students. Form an equation for a in terms of d.

b

At the beginning of the 13th year, the school had 649 students. Form another equation for a in terms of d.

c

Hence, solve for d, the number of students who joined the school each year.

d

How many students started at the school when it first opened?

e

At the beginning of the nth year, the school has reached its capacity at 1097 students. Solve for the value of n.

32

A diving vessel descends below the surface of the water at a constant rate so that the depth of the vessel after 4 minutes, 8 minutes and 12 minutes is 15 \text{ m}, 30 \text{ m} and 45 \text{ m} respectively.

If n is the number of minutes it takes to reach a depth of 120 \text{ m}, solve for n.

33

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?

n123418
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 line or an exponential curve?

34

The value of an investment that pays simple interest each year is graphed, where V_n is the value of the investment, in dollars, after n years.

a

State the value of V_1.

b

State the value of d, the amount of interest earned each year.

c

Use the results from the previous parts to write an explicit rule for V_n.

d

Find the value of the investment after 18 years.

1
2
3
4
5
n
500
1000
1500
2000
V_n
35

A car bought at the beginning of 2009 is worth \$1500 at the beginning of 2015. The value of the car has depreciated by a constant amount of \$50 each year since it was purchased.

a

Calculate the purchase price of the car in 2009.

b

Write an explicit rule for the value of the car after n years.

c

Solve for the year n at the end of which the car will be worth half the price it was bought for.

36

A piece of jewellery appreciates in value by a constant amount each year and its value is modelled by the recurrence relation:

V_n = V_{n - 1} + 320, \text{ } V_0 = 2000

where V_n is the value of the jewellery, in dollars, after n years.

a

State the initial value of the piece of jewellery.

b

By how much does it appreciate each year?

c

Write an explicit rule for V_n that gives the value of the piece of jewellery after n years.

d

What will the investment be worth after 11 years?

37

A motorbike depreciates in value by a constant amount each year and its value is modelled by the recurrence relation:

V_{n+1} = V_n - 1200, \text{ } V_0 = 15\,000

where V_{n+1} is the value of the motorbike, in dollars, after n+1 years.

a

State the purchase price of the motorbike.

b

At what rate is it decreasing in value each year?

c

Write an explicit rule for V_n that gives the value of the motorbike after n years.

d

What will the motorbike be worth after 9 years?

38

The balance of a savings account earning simple interest each year is given by the explicit rule V_n = 2200 + 300 \left(n - 1\right) where V_n is the balance after n years.

a

How much interest is the account earning each year?

b

Calculate the account balance after 1 year.

c

State the original investment amount.

d

Write a recursive rule for V_{n+1} in terms of V_{n}, and an initial condition V_0.

39

A mobile phone depreciates in value by a constant amount per month and its value is given by the explicit rule V_n = 1200 - 20 n where V_n is the balance after n months.

a

By how much does the value of the phone depreciate each month?

b

What was the purchase price of the phone?

c

Write a recursive rule for V_{n+1} in terms of V_n , and an initial condition V_0.

40

Zuber is a taxi service that charges a \$1.50 pick-up fee and \$1.95 per kilometre of travel.

a

Find the total charge for a 10\text{ km} journey.

b

Write a recursive rule for T_n in terms of T_{n - 1} which defines the price of a n\text{ km} trip, and an initial condition for T_0.

41

A rare figurine was purchased for \$60 and ten years later it is worth \$460.

a

How much did the figurine appreciate by each year if it appreciated in value by a constant amount each year?

b

Write a recursive rule for V_n in terms of V_{n - 1}, and an initial condition V_0.

c

Write an explicit rule, V_n, for the value of the figurine after n years.

d

Find the value of the figurine in another 10 years time.

42

A piece of machinery depreciated at a constant rate per hour of use. After 140 hours of use, it was worth \$28\,300. After 190 hours of use, it was worth \$28\,050.

a

Determine the amount of depreciation each hour.

b

State V_0, the initial value of the machinery.

c

Write a recursive rule for V_n in terms of V_{n - 1}, and an initial condition V_0.

d

Write an explicit rule, V_n, for the value of the machinery after n hours of use.

e

Solve for n, the number of hours of use after which the machinery will be worth a quarter of its original value.

43

Tabitha is a salesperson and is paid a travel allowance of \$45 for each business trip she takes to demonstrate the use of the product she sells. Each week she is also paid a retainer of \$230.

a

Calculate the total amount she is paid in a week where she does three business trips.

b

Write a recursive rule for u_n in terms of u_{n - 1} which defines how much Tabitha is paid in a week where she makes n business trips, and an initial condition for u_0.

c

Write a recursive rule for v_n in terms of v_{n - 1} which defines how much Tabitha is paid in two week where she makes n business trips, and an initial condition for v_0.

44

The value of Beirut Bank shares is decreasing by \$2.05 each day. At the beginning of today's trading, the shares are worth \$42.54.

a

Today is March 8. How much are they worth at the start of March 17?

b

Write a recursive rule for V_{n+1} in terms of V_n which defines the the value of the shares at the end of day n, and an initial condition for V_0.

45

Each day, Kumi withdraws \$4 from her bank account to spend on lunch. Before she withdraws this amount on January 1, she has \$2000 in her bank account.

a

After Kumi withdraws \$4 on January 7, calculate the amount left in her bank account.

b

How much does Kumi have left in her bank account at the end of February 4?

c

Write a recursive rule for T_{n} in terms of T_{n - 1} which defines the amount Kumi has in her account at the end of day n, and an initial condition for T_{0}.

46

When Alison was employed as a project manager, her initial salary was \$50\,000 per annum. After each year of service, she received a salary increase of \$2200.

a

Write an expression for T_{n}, the value of her salary in her nth year of working.

b

Calculate her salary in her 15th year of service.

c
When did her salary first exceed \$100\,000?
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Outcomes

MA12-2

models and solves problems and makes informed decisions about financial situations using mathematical reasoning and techniques

MA12-4

applies the concepts and techniques of arithmetic and geometric sequences and series in the solution of problems

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