1. Section 11.1 Sequences and Summation Notation Objectives: Definition and notation of sequences Recursively defined sequences Partial sums, including summation notation

2. Sequences A sequence is a set of numbers written in a specific order: a 1, a 2, a 3, a 4, …, a n, … – The number a 1 is called the first term, a 2 is the second term, and in general a n is the nth term. – Since for every natural number n, there is a corresponding number a n, we can define a sequence as a function.

3. Definition of a Sequence A sequence is a function f whose domain is the set of natural numbers. The values f(1), f(2), f(3),... are called the terms of the sequence.

4. Ex 1. Find the first 5 terms and the 100 th term of the sequence defined by each formula.

5. Class Work Find the first 5 terms and the 100 th term of the sequence defined by each formula.

6. Recursively Defined Sequences Some sequences do not have simple defining formulas like those of the preceding example. – The nth term of a sequence may depend on some or all of the terms preceding it. – A sequence defined in this way is called recursive.

7. Ex 2. Find the first 5 terms of the sequence defined recursively by a 1 = 1 and a n = 3(a n–1 + 2).

8. Class Work 3.Find the first 6 terms of the sequence defined recursively by a 1 = 3, a 2 = 8 and a n = a n-1 + a n-2.

9. The Partial Sums of a Sequence For the sequence a 1, a 2, a 3, a 4, …, a n, … the partial sums are: S 1 = a S 2 = a 1 + a 2 S 3 = a 1 + a 2 + a 3 S 4 = a 1 + a 2 + a 3 + a 4 S n = a 1 + a 2 + a 3 + … + a n

10. Ex 3. Find the first four partial sums of the sequence given by.

11. Class Work 4. Find the first four partial sums of the sequence given by.

12. Sigma Notation Given a sequence a 1, a 2, a 3, a 4,… we can write the sum of the first n terms using summation notation, or sigma notation. – This notation derives its name from the Greek letter Σ (capital sigma, corresponding to our S for “sum”).

13. Sigma notation is used as follows: – The left side of this expression is read: “The sum of a k from k = 1 to k = n.” – The letter k is called the index of summation, or the summation variable.

15. Ex 4. Find each sum.

16. Class Work Find each sum.

17. HW #1 Worksheet Sec 11.1