Download IB Math Studies Problem Set: Sequences/Series/Interest Name: 1

IB Math Studies
Problem Set: Sequences/Series/Interest
1.
Name:
________________________________
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)
2.
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)
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)
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)
IB Questionbank Mathematical Studies 3rd edition
1
4.
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)
5.
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)
6.
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)
IB Questionbank Mathematical Studies 3rd edition
2
7.
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)
(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)
8.
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)
IB Questionbank Mathematical Studies 3rd edition
3
9.
At what interest rate, compounded annually, would you need to invest $100 in order to have $125 in 2
years?
(Total 4 marks)
10. Astrid invests 1200 euros for five years at a nominal annual interest rate of 7.2 %, compounded
monthly.
(a)
Find the interest Astrid has earned during the five years of her investment.
Give your answer correct to two decimal places.
(3)
Helen invests 1200 euros in an annual simple interest scheme for five years.
She earns the same interest as Astrid.
(b)
Find the simple interest rate of this scheme.
(3)
(Total 6 marks)
11.
Daniel invests $1000 in an account that offers a nominal annual interest rate of 3.5 % compounded half
yearly.
(a)
Show that after three years Daniel will have $1109.70 in his account, correct to two decimal
places.
(3)
(b)
Write down the interest Daniel receives after three years.
(1)
Helen invests $1000 in an account that offers annual simple interest.
(c)
Find the annual simple interest rate that would give Helen $1109.70 after three years.
(3)
(Total 7 marks)
IB Questionbank Mathematical Studies 3rd edition
4