1. 11.1 Sequences & Summation Notation
Day Three
Revelation 3:20 "Here I am! I stand at the door and knock. If
anyone hears my voice and opens the door, I will come in and
eat with him, and he with me."

2. The sum of the first n terms of a sequence is
called the nth partial sum and is denoted Sn

3. The sum of the first n terms of a sequence is
called the nth partial sum and is denoted Sn
Find the indicated partial sums:
1) S10 for − 3, − 6, − 9, − 12, K

4. The sum of the first n terms of a sequence is
called the nth partial sum and is denoted Sn
Find the indicated partial sums:
1) S10 for − 3, − 6, − 9, − 12, K
−3 − 6 − 9 − 12 − 15 − 18 − 21− 24 − 27 − 30

5. The sum of the first n terms of a sequence is
called the nth partial sum and is denoted Sn
Find the indicated partial sums:
1) S10 for − 3, − 6, − 9, − 12, K
−3 − 6 − 9 − 12 − 15 − 18 − 21− 24 − 27 − 30
−165

6. Find the indicated partial sums:
2) S6 for an = 10n − 6

7. Find the indicated partial sums:
2) S6 for an = 10n − 6
4 + 14 + 24 + 34 + 44 + 54

8. Find the indicated partial sums:
2) S6 for an = 10n − 6
4 + 14 + 24 + 34 + 44 + 54
174

9. Find the indicated partial sums:
2) S6 for an = 10n − 6
4 + 14 + 24 + 34 + 44 + 54
174
A partial sum can be done on your calculator
as the sum of a sequence. Try it ...
sum ( seq (10x − 6, x, 1, 6, 1))

10. What is the syntax of this calculation?
sum ( seq (10x − 6, x, 1, 6, 1))

11. What is the syntax of this calculation?
sum ( seq (10x − 6, x, 1, 6, 1))
explicit formula
for the sequence

12. What is the syntax of this calculation?
sum ( seq (10x − 6, x, 1, 6, 1))
explicit formula
for the sequence
summation
variable

13. What is the syntax of this calculation?
sum ( seq (10x − 6, x, 1, 6, 1))
explicit formula
for the sequence
summation
variable
starts at

14. What is the syntax of this calculation?
sum ( seq (10x − 6, x, 1, 6, 1))
explicit formula
for the sequence
summation
variable
starts at ends at

15. What is the syntax of this calculation?
sum ( seq (10x − 6, x, 1, 6, 1))
explicit formula
for the sequence increases by
(called the step)
summation
variable
starts at ends at

16. Consider this: S1 , S2 , S3 , K Sn

17. Consider this: S1 , S2 , S3 , K Sn
It is a sequence of partial sums.
More on this later this chapter ...

18. Sigma Notation (or Summation Notation)

19. Sigma Notation (or Summation Notation)
n
∑a k = a1 + a2 + a3 +K + an
k=1

20. Sigma Notation (or Summation Notation)
n
∑a k = a1 + a2 + a3 +K + an
k=1
is read
“the sum of ak as k goes from 1 to n “

21. Sigma Notation (or Summation Notation)
n
∑a k = a1 + a2 + a3 +K + an
k=1
is read
“the sum of ak as k goes from 1 to n “
k is the summation variable ... or ...
index of summation

22. Sigma Notation (or Summation Notation)
n
∑a k = a1 + a2 + a3 +K + an
k=1
is read
“the sum of ak as k goes from 1 to n “
k is the summation variable ... or ...
index of summation
This is the math shorthand for doing the
sum of a sequence
just like what we did on the calculator!

23. Example:
Express 2 + 5 + 8 + 11+ 14 + 17 + 20 + 23 + 26
using Sigma Notation

24. Example:
Express 2 + 5 + 8 + 11+ 14 + 17 + 20 + 23 + 26
using Sigma Notation
Find the explicit expression for this sequence.
That will be your ak .
Then, write the Sigma Notation!

25. Example:
Express 2 + 5 + 8 + 11+ 14 + 17 + 20 + 23 + 26
using Sigma Notation
Find the explicit expression for this sequence.
That will be your ak .
Then, write the Sigma Notation!
9
∑ ( 3k − 1)
k=1

26. Example:
Express 2 + 5 + 8 + 11+ 14 + 17 + 20 + 23 + 26
using Sigma Notation
Find the explicit expression for this sequence.
That will be your ak .
Then, write the Sigma Notation!
9
∑ ( 3k − 1)
k=1
Practice calculating this on your calculator.

27. Example:
Express 2 + 5 + 8 + 11+ 14 + 17 + 20 + 23 + 26
using Sigma Notation
Find the explicit expression for this sequence.
That will be your ak .
Then, write the Sigma Notation!
9
∑ ( 3k − 1)
k=1
Practice calculating this on your calculator.
sum ( seq ( 3x − 1, x, 1, 9, 1)) 126

28. Find the sum by hand and verify with calculator:
8
1) ∑ ( 3k − 4 )
k=1

29. Find the sum by hand and verify with calculator:
8
1) ∑ ( 3k − 4 )
k=1
−1+ 2 + 5 + 8 + 11+ 14 + 17 + 20
76

30. Find the sum by hand and verify with calculator:
8
1) ∑ ( 3k − 4 )
k=1
−1+ 2 + 5 + 8 + 11+ 14 + 17 + 20
76
sum ( seq ( 3x − 4, x, 1, 8, 1))

31. Find the sum by hand and verify with calculator:
11
2) ∑4
k=7

32. Find the sum by hand and verify with calculator:
11
2) ∑4
k=7
4+4+4+4+4
20

33. Find the sum by hand and verify with calculator:
11
2) ∑4
k=7
4+4+4+4+4
20
sum ( seq ( 4, x, 7, 11, 1))
or
sum ( seq ( 4, x, 1, 5, 1))

34. Write this sum using Sigma Notation:
4 4 4 4 4 4 4 4
3) 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10

35. Write this sum using Sigma Notation:
4 4 4 4 4 4 4 4
3) 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10
10
4
∑k
k=3

36. Properties of Sums

37. Properties of Sums
n n n
1. ∑(a k + bk ) = ∑ ak + ∑ bk
k=1 k=1 k=1

38. Properties of Sums
n n n
1. ∑(a k + bk ) = ∑ ak + ∑ bk
k=1 k=1 k=1
n n n
2. ∑(a k − bk ) = ∑ ak − ∑ bk
k=1 k=1 k=1

39. Properties of Sums
n n n
1. ∑(a k + bk ) = ∑ ak + ∑ bk
k=1 k=1 k=1
n n n
2. ∑(a k − bk ) = ∑ ak − ∑ bk
k=1 k=1 k=1
n n
3. ∑(c ⋅ a ) = c ⋅ ∑ a
k k
k=1 k=1

40. HW #3
“We must not only give what we have, we must also
give what we are.” Desire Joseph Cardinal Mercier