11.1 An Introduction to Sequences & Series p. 651

What is a sequence? What is the difference between finite and infinite?

Sequence: A list of ordered numbers separated by commas.A list of ordered numbers separated by commas. Each number in the list is called a term.Each number in the list is called a term. For Example:For Example: Sequence 1 Sequence 2 2,4,6,8,10 2,4,6,8,10,… Term 1, 2, 3, 4, 5Term 1, 2, 3, 4, 5 Domain – relative position of each term (1,2,3,4,5) Usually begins with position 1 unless otherwise stated. Range – the actual terms of the sequence (2,4,6,8,10)

Sequence 1 Sequence 2 2,4,6,8,102,4,6,8,10,… 2,4,6,8,102,4,6,8,10,… A sequence can be finite or infinite. The sequence has a last term or final term. (such as seq. 1) The sequence continues without stopping. (such as seq. 2) Both sequences have a general rule: a n = 2n where n is the term # and a n is the nth term. The general rule can also be written in function notation: f(n) = 2n

Examples: Write the first 6 terms of a n =5-n.Write the first 6 terms of a n =5-n. a 1 =5-1=4a 1 =5-1=4 a 2 =5-2=3a 2 =5-2=3 a 3 =5-3=2a 3 =5-3=2 a 4 =5-4=1a 4 =5-4=1 a 5 =5-5=0a 5 =5-5=0 a 6 =5-6=-1a 6 =5-6=-1 4,3,2,1,0,-14,3,2,1,0,-1 Write the first 6 terms of a n =2 n.Write the first 6 terms of a n =2 n. a 1 =2 1 =2a 1 =2 1 =2 a 2 =2 2 =4a 2 =2 2 =4 a 3 =2 3 =8a 3 =2 3 =8 a 4 =2 4 =16a 4 =2 4 =16 a 5 =2 5 =32a 5 =2 5 =32 a 6 =2 6 =64a 6 =2 6 =64 2,4,8,16,32,642,4,8,16,32,64

Examples: Write a rule for the nth term. The seq. can be written as: Or, a n =2/(5 n ) The seq. can be written as:The seq. can be written as: 2(1)+1, 2(2)+1, 2(3)+1, 2(4)+1,… Or, a n =2n+1

Explicit Formula When the rules for a sequence are written so that the nth term can be calculated immediately, then it is expressed as an explicit formula. Examples: a n =2/(5 n ) or, a n =2n+1

Example: write an EXPLICIT rule for the nth term. 2,6,12,20,…2,6,12,20,… Can be written as:Can be written as: 1(2), 2(3), 3(4), 4(5),… Or, a n =n(n+1)

Recursive Form When given one or more of the initial terms and then defining the terms that follow using those previous terms is called a recursive formula. Example: find the 4 th term if

Try These … Find the sixth term of the sequences

A Famous Sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, … Found in flower petals, tree branches, bones in the human hand … THE FIBONACCI SEQUENCE

Convergent – Divergent Sequences If a sequence approaches a constant as the value of n gets large, the sequence is said to converge. If a sequence does NOT converge, it is divergent.

Divergent or Convergent? n a This sequence does not approach a constant … DIVERGENT

Divergent or Convergent? Hint: for explicit formulas, graph on your calculator and evaluate what happens when x gets large. For recursive formula, find the first several terms of the sequence and determine what will happen.

Divergent or Convergent? Try these …