1. The beginning
We are going to write a RECURRANCE RELATIONSHIP for the following sequence
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 …
RECURRANCE RELATIONSHIP is an equation describing which number comes next.
By Tom Bolan

2. 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 …
First we must understand what is happening:
1+1=2
Each number is the sum of the two before it
1+2=3
2+3=5
3+5=8
5+8=13

3. 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 …
The first term is 1
The second term is 1
The third term is 2
The seventh term is 13

4. 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89
The nth term is the sum of the 2 terms before it

5. 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89
This is a famous sequence (probably the only famous one) called the
“Fibonacci Sequence”
Believe it or not, this actually shows up in the real world.

6. Today we are going to learn about 2 things
SIGMA NOTATION
Shorthand for writing series
“Sigma”: Σ
Example:
LIMITS
The “target” of a sequence or series
What a function gets infinitely close to.

7. Sigma (Σ) notation 3 + 8 + 13 +…
Find the sum of the first 10 terms of the series …

8. Sigma (Σ) notation 3 + 8 + 13 +… The long way to do it:
Find the sum of the first 10 terms of the series …
The long way to do it:

9. Sigma (Σ) notation 3 + 8 + 13 +… The long way to do it:
Find the sum of the first 10 terms of the series …
The long way to do it:
But we know better, if n was 100, this would take forever, so we take a shortcut . . .

10. 3 + 8 + 13 +… + t10 Find the Explicit definition: tn = 3+5(n-1)
How much each term goes up by
The first term

11. 3 + 8 + 13 +… + t10 Find the Explicit definition: tn = 3+5(n-1)
Find the first and last terms
t1 = t10 = 48
We get this by putting in 10 for n

12. 3 + 8 + 13 +… + t10 Find the Explicit definition: tn = 3+5(n-1)
Find the first and last terms
t1 = t10 = 48
Use this formula
To find the sum of the first “n” terms.
“n”, in this case, is 10

13. 3 + 8 + 13 +… + t10 Find the Explicit definition: tn = 3+5(n-1)
Find the first and last terms
t1 = t10 = 48
Use the formula
4. Plug n’ chug
Put in 10 for n!

14. Term #1
Term #10

15. Along with a shorter way to solve it, there is a shorter way to write the question:

16. Instead of writing: Find the sum of the first 10 terms of the series 3 + 8 + 13 +…

17. Find the sum of the first 10 terms of the series 3 + 8 + 13 +…
Read: “the sum of 3+5(n-1) as n goes from 1 to 10 is 255.”

18. Stop here:
Start here:
Put 1 in for n

19. We put in 1-10
We got out this sequence
3,8,13,18,23,28,33,38,43,48
But this symbol means add ‘em up!

21. Write each of the following in sigma notation D

23. B

24. C

25. D

26. NOW FOR CALCULUS:

27. LIMITS
Find a partner
Find a location 8 squares away from a wall

28. LIMITS 1 partner will do the walking, the other will record
X=number of steps
Y = number of squares to the wall
At each step go half the distance to the wall
Record until y = 0 X Y 8 1 4 2

29. START WALKIN’

30. So how many steps does it take to reach the wall?4? ∞ 8?
64?

31. So you never actually reach the wall…
…but you get infinitely close!
As the number of steps you take
approaches infinity…
Your distance from the wall
approaches 0!

32. The sequence describing your distance to the wall…
8, 4, 2, 1, ½, …
Is described by the function:
Where “n” is the number of steps you take.

33. Distance = And as n (the number of steps) approaches infinity
Distance to the wall approaches 0.

34. “The limit, as n approaches infinity,
of is 0.”

35. Limits are the basic math behind calculus
The 2 basic things calculus is used to find are AREA and SLOPE!
But for really ugly functions

36. Pretend this is our function f(x)
And we want to know the slope right here
You can’t use rise/run because its curved!
But…

37. IF WE ZOOM IN…

42. A gazillion times

43. A gazillion times
Then it looks like a straight line!

44. A gazillion times Then it looks like a straight line!
When you do it an infinite number of times it is a straight line!

45. A gazillion times Then it looks like a straight line!
When you do it an infinite number of times it is a straight line!
So you can find the slope!

46. What if you want to find the area under the line?

47. You estimate, by finding the area of the rectangle.

48. If that estimate isn’t good enough, use more rectangles!

49. You guessed it…
An INFINITE number of rectangles!

50. CALCULUS IS . . .
Using the mathematics of infinity (limits), to make otherwise impossible calculations.
O.K., so I left a bunch of stuff out, but you get the basic idea.