1. Topic 1.1 – algebra arithmetic sequences & series

2. You Should Be Able To…
State whether a sequence is arithmetic, giving an appropriate reason
Find the common difference in an arithmetic sequence
Find the nth term of an arithmetic sequence
Find the number of terms in an arithmetic sequence
Solve real-world problems involving arithmetic sequences and series.
From IB SL Study Guide

3. How do you see this pattern growing?
Start with the title question and ask students to discuss in pairs.
Take responses from the class and record on the board the different ways students see the shapes growing.
Point to draw out: It is useful and natural that we see things different ways. This helps us represent math in different ways.
Instead of drawing shapes 4 and 5, students could build them using popsicle sticks or toothpicks.
The subsequent questions allow the class to generalise what they see.
Try to relate the way we see the growth to the algebraic representation.
This is the teaching time for the general term formula. Use the same notation as in the formula booklet. First term: u1, difference: d, term number: n
Draw shapes 4 and 5.
How many matchsticks are in shape 10?
Can you describe the pattern using algebra?

4. Each day a runner trains for a 10km race
Each day a runner trains for a 10km race. On the first day she runs 1,000m, and then increases the distance by 250m each subsequent day.
On which day does she run a distance of 10km in training?
10km = 10,000m and will be run on the 37th day.
A1 = 1000
an = (n-1)250 = 10,000
n = 37

5. In an arithmetic sequence, the first term is –2, the fourth term is 16, and the nth term is 11,998.
Find the common difference d.
Find the value of n
u4 = u1 + 3d
16 = -2 +3d
d = 6
11,998 = -2 + (n-1)6
n = 2001

6. Question – Finding Un Given Two Terms
In an arithmetic sequence, U7 = 121 and U15 = 193. Find the first three terms of the sequence and Un. Substitute know values in the formula for the nth term to write a system of equations. Then, solve the system. Since a = 67 and d = 9, the first three terms of the sequence are 67, 76, and 85.

7. Finding Un Given Two Terms continued…
To find Un , substitute 67 for a and 9 for d in the formula for the nth term. Un = 67 + (n – 1)9 Un = n – 9 Un = 9n + 58 Thus, the first three terms are 67, 76, and 85, and Un = 9n + 58.

8. You Should Know…
A sequence is arithmetic if the difference between consecutive terms is the same
An arithmetic sequence has the form:
u1, u1 + d, u1 + 2d, u1 + 3d, …, u1 + (n – 1)d
The common difference can be found by subtracting a term from the subsequent term:
d = un + 1 – un
When to use the term formula
Summary slide

9. You should know:

10. Arithmetic series

11. Arithmetic Series
Calculate the sum of the first n terms of an arithmetic series

12. Challenge
The top three layers of boxes in a store display are arranged as shown. If the pattern continues, and there are 12 layers in the display, what is the total number of boxes in the display?
Answer: 312

13. Sum of a Series Given First Terms
Find the sum of the first 60 terms of the series: (a) …

14. Sum of a Series Given First and Last Terms
Consider the series 17, 7, –3, …, –303. (a) Show that the series is arithmetic. Show that the difference between two consecutive terms is constant. For example: 7 – 17 = –3 – 7 = –10 Therefore, d = –10 and the series is arithmetic

15. Continued…
Consider the series 17, 7, –3, …, –303. (b) Find the sum of the series. The formula for the sum of an arithmetic series requires the value of n. Use the term formula first to find n. n = 33 Now use the appropriate formula to find the sum of the first 33 terms. S33 = –4719

16. Question
The sum of the first five terms of an arithmetic series is 65/2. Also, five times the 7th term is the same as six times the second term. Find the first term and common difference.

17. Question continued…

18. Be Prepared
Look for words or expressions that suggest the use of the term formula— “after the 10th month”, “in the 8th row”—and those that suggest the sum formula “total cost”, “total distance”, “altogether”.
Look for questions in which information is given about two terms. This normally suggests the formation of a pair of simultaneous equations that you will have to solve to find the first term and the common difference.
The last term of a sequence can be used to find the number of terms in the sequence

19. You should know:
When to use the sum formula