Lesson # ___ Section 8.1

Sequences & Summation Notation

You don’t have to write this down! Sequences are listed in order so that there is a first member, a second member, a third member and so on.. A sequence can also be thought of as a function whose domain is the set of positive integers {1,2,3,4,5….}

The numbers in a sequence are called TERMS Ex. Kelly has $31 in her account and deposits $10 each week for the next 4 weeks. The # of dollars forms a sequence: 31, 41, 51, 61, 71 The numbers in a sequence are called TERMS a1, a2, a3, a4, …an This is a finite sequence (it has a last term) a1, a2, a3, a4, … an … This is an infinite sequence (no last term)

Ex. Find the first 4 terms of the sequence given by an = (-1)n 2n -1

Finding the nth term of a sequence Write an expression for the nth term (an) of each sequence. b. 0, 3, 8, 15, 24… c. 1/2, -1/4, 1/8, -1/16,… 𝒏 1 2 3 4 5 𝒂 𝒏 𝒂 𝒏 =

We can also define sequences recursively 𝒏 1 2 3 4 𝒂 𝒏 5 7 9 Most sequences are defined explicitly, for example 𝒂 𝒏 =𝟐𝒏+𝟏. Here, the nth term is defined in terms of n. Using the language of functions, we could say that the output is defined in terms of the input. If you want the 100th term, simply plug in n = 100. We can also define sequences recursively for example 𝒂 𝟏 =𝟑, 𝒂 𝒏 = 𝒂 𝒏−𝟏 +𝟐 Here, the nth term is defined in terms of previous terms of the sequence. Using the language of functions, we could say that the output is defined in terms of it’s relationship to previous outputs. If you want the 100th term, you’d simply add 2 to the 99th term, which is 2 more than the 98th term, and so on. Note that we need to also give an initial value in order to define a sequence this way.

1, 1, 2, ___, ___, ___, ___ a0 = 1, a1 = 1, an = an-2 + an-1 The Fibonacci Sequence is an example of a sequence that is defined recursively a0 = 1, a1 = 1, an = an-2 + an-1 1, 1, 2, ___, ___, ___, ___

6 ! = 6 * 5 * 4 * 3 * 2 * 1 Factorial Notation n ! = n * (n-1) * (n-2) * … * 3 * 2 * 1 Note: 0 ! = 1 by definition Simplify 8! 5! 3! Simplify 𝑛! 𝑛−2 !

Summation (or Sigma) Notation See the definition on p 584 These are like algebraic adding machines. They produce a whole sequence of numbers and add them up. 𝑖=1 4 2𝑥 =2 1 2 2 + + + 2 3 2 4 𝑖=1 3 (𝑥+2) =

And whatever you do, don’t forget to give lots of homework! Hey, Mr. B don’t forget to mention the properties on p 585! I love this job!