1. Sequences and Summations

2. Sequences

3. Definition of a sequence
A sequence is a function from a subset of integers to a set S.

4. An example of a sequence
Beginning with n=1

5. A geometric progression
Beginning with n=0

6. An arithmetic progression
Beginning with n=0

7. Example progression

8. Summations

9. Summations
Sigma for “sum”

10. Product
Pi for “product”

11. Summations
Upper limit
Lower limit
Index of summation

12. Summations example
The sum of the first 100 integers
What’s the answer?

13. Summations example/picture
We now have a square 10 (n) by 11 (n+1) with area 110 units
We need half of that (10x11)/2

14. Summations challenge

15. Summations challenge

16. Summations: shifting indices

17. Summations: a theorem
Proof: see the book

18. Summations: nesting

19. Summations: implementing a nesting as a bit of program
for i = 1 to n-1 do
for j=i+1 to n do
sum = sum + i*j

20. Summations: just a wee tweak

21. One More Example

22. One More Example
Table 2, page 157

23. One More Example
Then use this

24. Challenge