Alg2 8.1 Defining and Using Sequences and Series
Sequences
A sequence is an ordered list of numbers. A finite sequence is a function that has a limited number of terms
and whose domain is the finite set {1, 2, 3, … , 𝑛}. The values in the range are called the terms of the sequence.
Finite sequence: 2, 5, 8, 11, 14, 17, 20
Rule: 𝑎𝑛 = 3𝑛 − 1
Infinite sequence: 2, 5, 8, 11, 14, 17, 20, …
Ex#1:
Ex#2:
Write the first six terms of each sequence:
1
a)
𝑎𝑛 = 𝑛 − 3
2
b)
Write a rule for the nth term of each sequence:
a) 5, −15, 45, −135, 405, …
b)
c)
9, 16, 25, 36, 49, …
e)
√2, √2√2, √2√2√2, √2√2√2√2, …
d)
𝑓(𝑛) = (−1)𝑛 + 3
20, 16, 12, 8, 4, …
1 7 1 7 1
, , , ,
,…
1 2 6 24 120
Series and Summation Notation
When the terms of a sequence are added together, the resulting expression is a series. A series can be finite or
infinite and can be represented using summation notation (also called sigma notation).
4
4 + 7 + 10 + 13 = ∑ 3𝑛 + 1
𝑖=1
Ex#3:
Ex#4:
Write each series using summation notation.
a) −1 + 2 + (−3) + 4 + ⋯ + 40
b)
1 1 1
1+ + +
+⋯
4 9 16
100
1
1
∑( −
)=
𝑖 𝑖+1
𝑖=1
Properties of Sums
𝑛
𝑛
𝑛
𝑛
𝑛
𝑛
𝑛
𝑛
∑ 𝑐𝑎𝑖 = 𝑐 ∑ 𝑎𝑖
∑(𝑎𝑖 + 𝑏𝑖 ) = ∑ 𝑎𝑖 + ∑ 𝑏𝑖
∑(𝑎𝑖 − 𝑏𝑖 ) = ∑ 𝑎𝑖 − ∑ 𝑏𝑖
𝑖=1
𝑖=1
𝑖=1
𝑖=1
Formulas for Special Series
𝑛
𝑛(𝑛 + 1)
∑𝑖 = 1 + 2 + ⋯+ 𝑛 =
2
𝑖=1
𝑛
∑ 𝑖 2 = 12 + 22 + ⋯ + 𝑛 2 =
𝑖=1
Ex#5:
𝑖=1
𝑖=1
𝑛
∑ 𝑖 3 = 13 + 23 + ⋯ + 𝑛 3 =
𝑖=1
𝑖=1
𝑖=1
𝑛2 (𝑛 + 1)2
4
𝑛(𝑛 + 1)(2𝑛 + 1)
6
100
∑ 5𝑖 3 =
𝑖=50
HW: Alg2 8.1 p.414-416 #6-15 (x3), #16, #22-26, #27-48 (x3), #53, #61 & PC 8.1 p.588 #50-56 (even)