# Lesson 10.1: Sequences & Summation Notation

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Lesson 10.1: Sequences & Summation Notation
Algebraic formula first n terms Factorial notation Series Summation or Sigma notation

Sequence: an ordered collection of comma-separated terms
, 9, 13, 17, _____ , 1, -1, 1, -1, 1, _____ , 1, 2, 3, 5, 8, 13, 21, ______ , , 100, - 10, 1, ______ a + b, 2b, - a + 3b, - 2a + 4b, _________

An infinite sequence { an } is a function whose domain
is the set of positive integers*. The function values a1, a2, a3, a4, ... an, ... are the terms of the sequence. If the domain of the sequence consists of only the first n positive integers, then the sequence is a finite sequence. [p 656] Example: infinite sequence 5, 9, 13, 17, , an, . . . an nth term a1 first term What is a6? ________ Occasionally, sequences start with n=0

Algebraic formula first n terms
Find the first four terms of the indicated sequence. an = 3n – 2 a1 = 3(1) – 2 = sequence: a2 = 3(2) – 2 = a3 = 3(3) – 2 = a4 =

Find the first four terms of the indicated sequence.

Practice: p # 7 & 10 Write the first five terms of the indicated sequence.

Algebraic formula first n terms
Write an expression for the apparent nth term of the sequence , 3, 5, 7, . . . n: n terms: an Answer: an = 2n - 1

Write an expression for the apparent nth term of the sequence
Answer: , 5, 10, 17, . . . n: n terms: an Answer:

Factorial Notation [p 658]
If n is a positive integer, then n factorial is defined by n! = 1 · 2 · 3 · 4 · 5 · · · (n – 1) · n Special case: 0! = 1 4! = 1 · 2 · 3 · 4 = 24 10! = 1 · 2 · 3 · 4 · 5 · · ·10 = 3,628,800 On graphing calculators: On Ti89: MATH Probability ! On Ti84: MATH PRB !

Find the first five terms of the sequence whose nth
term is Begin with n = 0. sequence:

Fractions involving factorials can often be simplified.
1. 2. 3. 4.

Series: the summation of the terms of a sequence
The sum of the first n terms of a sequence is represented by where i is the index of summation, n is the upper limit of summation and 1 is the lower limit of summation*. [p 660] This is often referred to as sigma notation. The index does not have to i and the lower index of summation does not have to be 1.

Properties of Sums 4(1) + 4(2) + 4(3) + 4(4) + 4(5) = 60
c is any constant (22 + 2) + (32 + 2) + (42 + 2) + (52 + 2) + (62 + 2) = 100

Use Sigma Notation to write a sum
Similar to writing on expression for the apparent nth term. Place in sigma notation, specifying summation boundaries. n: n terms: an an = 3n - 2 – – n: n terms: – – –729 an

Homework: Sequences: p664 #2, 4, 6, 12, 14, 16, 22, 26, 28, 30, 34 Series: pp 664–665 #38, 40, 42, 46, 48, 52, 55