1.
A teacher earns an annual salary of 45 000 USD for the first year of her employment
Her annual salary increases by 1750 USD each year.
(a)
Calculate the annual salary for the fifth year of her employment.
(3)
She remains in this employment for 10 years.
(b)
Calculate the total salary she earns in this employment during these 10 years.
(3)
(Total 6 marks)
2.
Consider the arithmetic sequence 1, 4, 7, 10, 13, …
(a)
Find the value of the eleventh term.
(2)
(b)
The sum of the first n terms of this sequence is
n
(3n – 1).
2
(i)
Find the sum of the first 100 terms in this arithmetic sequence.
(ii)
The sum of the first n terms is 477.
(a)
Show that 3n2 – n – 954 = 0.
(b)
Using your graphic display calculator or otherwise, find the number of
terms, n.
(6)
(Total 8 marks)
3.
A concert choir is arranged, per row, according to an arithmetic sequence. There are 20 singers
in the fourth row and 32 singers in the eighth row.
(a)
Find the common difference of this arithmetic sequence.
(3)
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There are 10 rows in the choir and 11 singers in the first row.
(b)
Find the total number of singers in the choir.
(3)
(Total 6 marks)
4.
An arithmetic sequence is defined as
un = 135 + 7n,
(a)
n = 1, 2, 3, …
Calculate u1, the first term in the sequence.
(2)
(b)
Show that the common difference is 7.
(2)
Sn is the sum of the first n terms of the sequence.
(c)
Find an expression for Sn. Give your answer in the form Sn = An2 + Bn, where A and B are
constants.
(3)
The first term, v1, of a geometric sequence is 20 and its fourth term v4 is 67.5.
(d)
Show that the common ratio, r, of the geometric sequence is 1.5.
(2)
Tn is the sum of the first n terms of the geometric sequence.
(e)
Calculate T7, the sum of the first seven terms of the geometric sequence.
(2)
(f)
Use your graphic display calculator to find the smallest value of n for which Tn > Sn.
(2)
(Total 13 marks)
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5.
The first term of an arithmetic sequence is 3 and the sum of the first two terms is 11.
(a)
Write down the second term of this sequence.
(1)
(b)
Write down the common difference of this sequence.
(1)
(c)
Write down the fourth term of this sequence.
(1)
(d)
The nth term is the first term in this sequence greater than 1000.
Find the value of n.
(3)
(Total 6 marks)
6.
A tree begins losing its leaves in October. The number of leaves that the tree loses each day
increases by the same number on each successive day.
Date in October
1
2
3
4
.....................
Number of leaves lost
24
40
56
72
.....................
(a)
Calculate the number of leaves that the tree loses on the 21st October.
(3)
(b)
Find the total number of leaves that the tree loses in the 31 days of the month of October.
(3)
(Total 6 marks)
7.
Consider the following sequence:
57, 55, 53 . . . , 5, 3
(a)
Find the number of terms of the sequence.
(3)
(b)
Find the sum of the sequence.
(3)
(Total 6 marks)
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8.
The first term of an arithmetic sequence is 7 and the sixth term is 22. Find
(a)
the common difference;
(2)
(b)
the twelfth term;
(2)
(c)
the sum of the first 100 terms.
(2)
(Total 6 marks)
9.
The first term of an arithmetic sequence is 0 and the common difference is 12.
(a)
Find the value of the 96th term of the sequence.
(2)
The first term of a geometric sequence is 6. The 6th term of the geometric sequence is equal to
the 17th term of the arithmetic sequence given above.
(b)
Write down an equation using this information.
(2)
(c)
Calculate the common ratio of the geometric sequence.
(2)
(Total 6 marks)
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10.
Give all answers in this question correct to the nearest dollar.
Clara wants to buy some land. She can choose between two different payment options.
Both options require her to pay for the land in 20 monthly installments.
Option 1:
The first installment is $2500. Each installment is $200 more than the one before.
Option 2:
The first installment is $2000. Each installment is 8 more than the one before.
(a)
If Clara chooses option 1,
(i)
write down the values of the second and third installments;
(ii)
calculate the value of the final installment;
(iii)
show that the total amount that Clara would pay for the land is $88 000.
(7)
(b)
If Clara chooses option 2,
(i)
find the value of the second installment;
(ii)
show that the value of the fifth installment is $2721.
(4)
(c)
The price of the land is $80 000. In option 1 her total repayments are $88 000 over the 20
months. Find the annual rate of simple interest that gives this total.
(4)
(d)
Clara knows that the total amount she would pay for the land is not the same for both
options. She wants to spend the least amount of money. Find how much she will save by
choosing the cheaper option.
(4)
(Total 19 marks)
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11.
Given the arithmetic sequence: u1 = 124, u2 = 117, u = 110, u4 = 103, …
(a)
Write down the common difference of the sequence.
(1)
(b)
Calculate the sum of the first 50 terms of the sequence.
(2)
uk is the first term in the sequence that is negative.
(c)
Find the value of k.
(3)
(Total 6 marks)
12.
Throughout this question all the numerical answers must be given correct to the nearest
whole number.
Park School started in January 2000 with 100 students. Every full year, there is an increase of
6 % in the number of students.
(a)
Find the number of students attending Park School in
(i)
January 2001;
(ii)
January 2003.
(4)
(b)
Show that the number of students attending Park School in January 2007 is 150.
(2)
Grove School had 110 students in January 2000. Every full year, the number of students is 10
more than in the previous year.
(c)
Find the number of students attending Grove School in January 2003.
(2)
(d)
Find the year in which the number of students attending Grove School will be first 60 %
more than in January 2000.
(4)
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Each January, one of these two schools, the one that has more students, is given extra money to
spend on sports equipment.
(e)
(i)
Decide which school gets the money in 2007. Justify your answer.
(ii)
Find the first year in which Park School will be given this extra money.
(5)
(Total 17 marks)
13.
The natural numbers: 1, 2, 3, 4, 5… form an arithmetic sequence.
(a)
State the values of u1 and d for this sequence.
(2)
(b)
Use an appropriate formula to show that the sum of the natural numbers from 1 to n is
1
given by
n (n +1).
2
(2)
(c)
Calculate the sum of the natural numbers from 1 to 200.
(2)
(Total 6 marks)
14.
The fifth term of an arithmetic sequence is 20 and the twelfth term is 41.
(a)
(i)
Find the common difference.
(2)
(ii)
Find the first term of the sequence.
(1)
(b)
Calculate the eighty-fourth term.
(1)
(c)
Calculate the sum of the first 200 terms.
(2)
(Total 6 marks)
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D ô15. The first three terms of an arithmetic sequence are
2k + 3, 5k − 2 and 10k −15.
(a)
Show that k = 4.
(3)
(b)
Find the values of the first three terms of the sequence.
(1)
(c)
Write down the value of the common difference.
(1)
(d)
Calculate the 20th term of the sequence.
(2)
(e)
Find the sum of the first 15 terms of the sequence.
(2)
(Total 9 marks)
16.
Two students Ann and Ben play a game. Each time Ann passes GO she receives $15. Each time
Ben passes GO he receives 8% of the amount he already has. Both students start with $100.
(a)
How much money will Ann have after she has passed GO 10 times?
(b)
How much money will Ben have after he passes GO 10 times?
(c)
How many times will the students have to pass GO for Ben to have more money than
Ann?
(Total 6 marks)
17.
The first five terms of an arithmetic sequence are shown below.
2,
6,
10,
14,
18
(a)
Write down the sixth number in the sequence.
(b)
Calculate the 200th term.
(c)
Calculate the sum of the first 90 terms of the sequence.
(Total 8 marks)
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18.
(a)
The first term of an arithmetic sequence is –16 and the eleventh term is 39.
Calculate the value of the common difference.
(b)
The third term of a geometric sequence is 12 and the fifth term is
16
.
3
All the terms in the sequence are positive.
Calculate the value of the common ratio.
(Total 8 marks)
19.
The nth term of an arithmetic sequence is given by un = 63 – 4n.
(a)
Calculate the values of the first two terms of this sequence.
(2)
(b)
Which term of the sequence is –13?
(2)
(c)
Two consecutive terms of this sequence, uk and uk + 1, have a sum of 34. Find k.
(3)
(Total 7 marks)
20.
The sixth term of an arithmetic sequence is 24. The common difference is 8.
(a)
Calculate the first term of the sequence.
The sum of the first n terms is 600.
(b)
Calculate the value of n.
(Total 8 marks)
21.
The fourth term of an arithmetic sequence is 12 and the tenth term is 42.
(a)
Given that the first term is u1 and the common difference is d, write down two equations
in u1 and d that satisfy this information.
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(b)
Solve the equations to find the values of u1 and d.
(Total 8 marks)
22.
Ann and John go to a swimming pool.
They both swim the first length of the pool in 2 minutes.
The time John takes to swim a length is 6 seconds more than he took to swim the previous
length.
The time Ann takes to swim a length is 1.05 times that she took to swim the previous length.
(a)
(i)
Find the time John takes to swim the third length.
(ii)
Show that Ann takes 2.205 minutes to swim the third length.
(3)
(b)
Find the time taken for Ann to swim a total of 10 lengths of the pool.
(3)
(Total 6 marks)
23.
A National Lottery is offering prizes in a new competition. The winner may choose one of the
following.
Option one:
$1000 each week for 10 weeks.
Option two:
$250 in the first week, $450 in the second week, $650 in the third week,
increasing by $200 each week for a total of 10 weeks.
Option three:
$10 in the first week, $20 in the second week, $40 in the third week
continuing to double for a total of 10 weeks.
(a)
Calculate the amount you receive in the tenth week, if you select
(i)
option two;
(ii)
option three.
(6)
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(b)
What is the total amount you receive if you select option two?
(2)
(c)
Which option has the greatest total value? Justify your answer by showing all appropriate
calculations.
(4)
(Total 12 marks)
24.
On Vera’s 18th birthday she was given an allowance from her parents. She was given the
following choices.
Choice A
Choice B
Choice C
Choice D
(a)
$100 every month of the year.
A fixed amount of $1100 at the beginning of the year, to be invested at an interest
rate of 12% per annum, compounded monthly.
$75 the first month and an increase of $5 every month thereafter.
$80 the first month and an increase of 5% every month.
Assuming that Vera does not spend any of her allowance during the year, calculate, for
each of the choices, how much money she would have at the end of the year.
(8)
(b)
Which of the choices do you think that Vera should choose? Give a reason for your
answer.
(2)
(c)
On her 19th birthday Vera invests $1200 in a bank that pays interest at r% per annum
compounded annually. Vera would like to buy a scooter costing $1452 on her
21st birthday. What rate will the bank have to offer her to enable her to buy the scooter?
(4)
(Total 14 marks)
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25.
The first four terms of an arithmetic sequence are shown below.
1, 5, 9, 13,......
(a)
Write down the nth term of the sequence.
(b)
Calculate the 100th term of the sequence.
(c)
Find the sum of the first 100 terms of the sequence.
(Total 4 marks)
26.
Mr Jones decides to increase the amount of money he spends on food by d GBP every year. In
the first year he spends a GBP. In the 8th year he spends twice as much as in the 4th year. In the
20th year he spends 4000 GBP.
Find the value of d.
(Total 4 marks)
27.
A woman deposits $100 into her son’s savings account on his first birthday. On his second
birthday she deposits $125, $150 on his third birthday, and so on.
(a)
How much money would she deposit into her son’s account on his 17th birthday?
(b)
How much in total would she have deposited after her son’s 17th birthday?
(Total 4 marks)
28.
The seventh term, u7, of a geometric sequence is 108. The eighth term, u8, of the sequence is 36.
(a)
Write down the common ratio of the sequence.
(1)
(b)
Find u1.
(2)
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The sum of the first k terms in the sequence is 118 096.
(c)
Find the value of k.
(3)
(Total 6 marks)
29.
A geometric sequence has 1024 as its first term and 128 as its fourth term.
(a)
Show that the common ratio is
1
.
2
(2)
(b)
Find the value of the eleventh term.
(2)
(c)
Find the sum of the first eight terms.
(3)
(d)
Find the number of terms in the sequence for which the sum first exceeds 2047.968.
(3)
(Total 10 marks)
30.
Consider the geometric sequence 16, 8, a, 2, b, …
(a)
Write down the common ratio.
(1)
(b)
Write down the value of
(i)
a;
(ii)
b.
(2)
(c)
The sum of the first n terms is 31.9375. Find the value of n.
(3)
(Total 6 marks)
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31.
The annual fees paid to a school for the school years 2000, 2001 and 2002 increase as a
geometric progression. The table below shows the fee structure.
(a)
Year
Fees (USD)
2000
8000.00
2001
8320.00
2002
8652.80
Calculate the common ratio for the increasing sequence of fees.
(2)
In parts (b) and (c) give your answer correct to 2 decimal places.
The fees continue to increase in the same ratio.
(b)
Find the fees paid for 2006.
(2)
A student attends the school for eight years, starting in 2000.
(c)
Find the total fees paid for these eight years.
(2)
(Total 6 marks)
32.
A geometric sequence has second term 12 and fifth term 324.
(a)
Calculate the value of the common ratio.
(4)
(b)
Calculate the 10th term of this sequence.
(3)
(c)
The kth term is the first term that is greater than 2000. Find the value of k.
(3)
(Total 10 marks)
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33.
The population of big cats in Africa is increasing at a rate of 5 % per year. At the beginning of
2004 the population was 10 000.
(a)
Write down the population of big cats at the beginning of 2005.
(1)
(b)
Find the population of big cats at the beginning of 2010.
(2)
(c)
Find the number of years, from the beginning of 2004, it will take the population of big
cats to exceed 50 000.
(3)
(Total 6 marks)
34.
A geometric progression G1 has 1 as its first term and 3 as its common ratio.
(a)
The sum of the first n terms of G1 is 29 524. Find n.
(3)
A second geometric progression G2 has the form 1,
(b)
1 1 1
, ,
…
3 9 27
State the common ratio for G2.
(1)
(c)
Calculate the sum of the first 10 terms of G2.
(2)
(d)
Explain why the sum of the first 1000 terms of G2 will give the same answer as the sum
of the first 10 terms, when corrected to three significant figures.
(1)
(e)
Using your results from parts (a) to (c), or otherwise, calculate the sum of the first 10
1
1
1
terms of the sequence 2, 3 , 9 , 27
…
3
9
27
Give your answer correct to one decimal place.
(3)
(Total 10 marks)
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35.
Annie is starting her first job. She will earn a salary of $26000 in the first year and her salary
will increase by 3 every year.
(a)
Calculate how much Annie will earn in her 5th year of work.
(3)
Annie spends $24800 of her earnings in her first year of work. For the next few years, inflation
will cause Annie’s living expenses to rise by 5 per year.
(b)
(i)
Calculate the number of years it will be before Annie is spending more than she
earns.
(ii)
By how much will Annie’s spending be greater than her earnings
in that year?
(6)
(Total 9 marks)
36.
Consider the geometric sequence 8, a, 2,… for which the common ratio is
(a)
Find the value of a.
(b)
Find the value of the eighth term.
(c)
Find the sum of the first twelve terms.
1
.
2
(Total 6 marks)
37.
A geometric sequence has all its terms positive. The first term is 7 and the third term is 28.
(a)
Find the common ratio.
(b)
Find the sum of the first 14 terms.
(Total 6 marks)
38.
A basketball is dropped vertically. It reaches a height of 2 m on the first bounce. The height of
each subsequent bounce is 90% of the previous bounce.
(a)
What height does it reach on the 8th bounce?
(2)
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(b)
What is the total vertical distance travelled by the ball between the first and sixth time the
ball hits the ground?
(4)
(Total 6 marks)
39.
The tuition fees for the first three years of high school are given in the table below.
Year
Tuition fees
(in dollars)
1
2000
2
2500
3
3125
These tuition fees form a geometric sequence.
(a)
Find the common ratio, r, for this sequence.
(b)
If fees continue to rise at the same rate, calculate (to the nearest dollar) the total cost of
tuition fees for the first six years of high school.
(Total 4 marks)
40.
The population of Bangor is growing each year. At the end of 1996, the population was 40 000.
At the end of 1998, the population was 44 100. Assuming that these annual figures follow a
geometric progression, calculate
(a)
the population of Bangor at the end of 1997;
(b)
the population of Bangor at the end of 1992.
(Total 4 marks)
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