IB Math Studies
Topic 2: Sequences and Series
You should be able to do the following things on the test:
Find the next term of any arithmetic or geometric sequence
Find the common difference of an arithmetic sequence
Find the common ratio of a geometric sequence
Find the nth term of any arithmetic or geometric sequence
Find how many terms there are in an arithmetic or geometric sequence
Create a specific number of terms between terms in any arithmetic sequence
Create a specific number of terms between terms in any geometric sequence
Find the sum of an arithmetic or geometric sequence/series
Find anarithmetic or geometric sum written in sigma Σ notation
Solve applied problems using arithmetic or geometric sequences and series
1.A woman deposits $100 into her son’s savings account on his first birthday. On his second birthday she deposits $125. On his third birthday, she deposits $150, 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?
2.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.
3. In 2000 Herman joined a tennis club. The fees were £ 1200 a year. Each year the fees increase by 3%.
a) Calculate, to the nearest £ 1, the fees in 2002.
b) Calculate the total fees for Herman who joined the tennis club in 2000 and remained amember for five years.
4.The first term of an arithmetic sequence is -16 and the eleventh term is 39. Calculate the common difference.
IB Math Studies
Arithmetic Sequences
A small business sells $10,000 worth of products during its first year. The owner of the business has set a goal of increasing annual sales by $2000 each year for the next 9 years.
a)Find the sales in the 4th year and the 10th year.
b)Graph the sales for each year.
c)What kind of function models this situation?
An arithmetic sequence is a list of numbers with a common difference between each successive term
–5 , 7 , 19 , ____ , ____ , ____ , …
r + 15 , r + 8 , r + 1, ______, ______, ______, …
Find the 20th term of –11, –2, 7,…
The general, or nth, term of an arithmetic sequence is found by ______.
Why?
Examples:
- Find the 100th term in the sequence 1, 5, 9, 13, 17, …
- Find the formula for the nth term in the sequence 16, 7, –2, …
- Given the sequence find.
- For the arithmetic sequence find the first term.
- How many terms are in the sequence 3, 9, 15, …, 81, 87?
Try These:
Consider the sequence 6 , 17 , 28 , 39 , …, 897, 908.
a)Show that the sequence is arithmetic.
b)Find the formula for its general term.
c)Find its 50th term.
d)How many terms are in the sequence?
e)Is 325 a member of the sequence? How do you know?
f)Is 761 a member of the sequence? How do you know?
IB Math Studies
Arithmetic Series and Sigma Notation
A free-falling skydiver speeds up as she falls. That means that each second, she falls a longer distance than the precious second. She falls 16 feet during the first second, 48 feet during the next second, 80 feet during the third second, and so on.
How far will she fall during the 4th second? During the fifth second?
How far will she fall during all five seconds?
An arithmetic series is the sum of the terms in an arithmetic sequence.
There are two formulas that can be used to find the sum of an arithmetic series:
Examples:
- Find the sum of the first 13 terms in the sequence –5, 1, 7, …
- Find the sum of the terms in the series –14 – 8 – 2 + … + 142
3.Ron agrees with his son Brett’s request for an increase in his allowance of $0.75 per week for 24 weeks. Brett’s allowance now is $3.00.
- How much will his allowance be on the 24th week?
- What will be the total amount of allowance that Brett has received over 24 weeks?
Try These
a)Find the sum of the first 40 terms of 100 + 93 + 86 + 79 + …
b)Find the sum of 35000, 36750, 38500, …, 68250
Sigma Notation
The Greek letter sigma Σ is often used to express the concept of adding up a series of terms. The Σnotation is used in the following form:
The notation is read as the sum of all the u terms from i=1 to i=n where:
- represents the term that are being added
- is the variable that is used to count to the next term
- is the last value of to be counted
Find all the terms Use the Sn formula
IB Math Studies
Geometric Sequences
For his science experiment, Ahad exposed a 100 mm plant to a special light for 20 hours each day. He found that the plants grew about 10% each month.
a)How tall will the plant be at the beginning of the eighth month?
b)Graph the height at the beginning of each month.
1 2 3 4 5 6 7 8 9 10
A geometric sequence is a list of numbers with a common ratio between each successive term
5 , 15 , 45 , ____ , ____ , ____ , …
6 , –24 , 96 , ______, ______, ______, …
81 , 27 , 9 , ____ , ____ , ____ , …
1280 , 960 , 720 , ____ , ____ , ____ , …
15 , 30 , 45 , ______, ______, ______, …
The general, or nth, term of a geometric sequence is found by ______.
Why?
Examples:
- Find the formula for the nth (general) term in the sequence 3, 12, 48… What kind of function is this? Find the 11th term.
- Find the first term in the sequence for which = 24 and r = 2.
- Find a geometric sequence that has 2 terms between –2 and 54.
- A colony of algae increases its size by 15% each week. The lake will be considered “seriously polluted” when there is an excess of 10,000 grams of algae in the lake. If 10 grams are placed in the lake, how long will it take for the lake to be seriously polluted?
Try These:
Consider the sequence 100 , 50 , 25 , …
a)Show that the sequence is geometric.
b)Find the formula for its general term. Find its 15th term.
c)Is 7 a member of the sequence? How do you know?
e)Is a member of the sequence? How do you know?
IB Math Studies
Geometric Series
A geometric seriesis the sum of terms in a geometric sequence.
The general formula for the sum of the first n terms of a geometric sequence is
______or .
Examples:
- Find the sum of the first 8 terms in the sequence 3 – 6 + 12 – …
- Find the sum of the terms in the series
Try These
- Find the sum of the first ten terms of 12 + 6 + 3 + …
- An employee of a company starts on a salary of $20,000 per year with an annual increase of 4% of the previous year’s salary
a)Show that the amounts of the salary form a geometric sequence
b)Find how much the employee earns in the tenth year of employment.
c)Find the total amount the employee makes over all 10 years.
3.A National Lottery is offering prizes in a new competition. The winner maychoose 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 thirdweek, 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 thirdweek continuing to double for a total of 10 weeks.
a)Calculate the amount you receive in the tenth week for each option.
b) Calculate the total amount you receive over all ten weeks for each option.
Sigma Notation
The Greek letter sigma Σ is often used to express the concept of adding up a series of terms. The Σnotation is used in the following form:
The notation is read as the sum of all the u terms from i=1 to i=n where:
- represents the term that are being added
- is the variable that is used to count to the next term
- is the last value of to be counted
Find all the terms Use the Sn formula