GPS – Sequences and Series  MA3A9. Students will use sequences and series  a. Use and find recursive and explicit formulae for the terms of sequences.  b. Recognize and use simple arithmetic and geometric sequences.  c. Investigate limits of sequences.  d. Use mathematical induction to find and prove formulae for sums of finite series.  e. Find and apply the sums of finite and, where appropriate, infinite arithmetic and geometric series.  f. Use summation notation to explore series.  g. Determine geometric series and their limits.

Chapter 14 Sequences and Series

14-1 : Introduction to Sequences and Series  Sequence – 1) an ordered list of numbers. 2) a function whose domain is the set of positive integers.  Series – the sum of the numbers in a sequence.  Finite Sequence – has a countable number of terms.  Infinite Sequence – has an uncountable number of terms.

14-1 : Introduction to Sequences and Series  Describe the pattern and find the next three terms.  2,4,6,8,__,__,__  5,2,-1,-4,__,__,__  3,6,12,24,__,__,__  1,4,9,16,__,__,__  1,3/2,5/3,7/4,__,__,__

14-1 : Introduction to Sequences and Series  Recursive Formula – A formula for terms of a sequence that specifies each term as a function of the preceding term(s).  Explicit Formula – A formula for terms of a sequence that specifies each term as a function of n (the number of the specified term)

14-2 : Arithmetic, Geometric, and Other Sequences  Discrete Function – A function whose domain is a set of disconnected values.  Continuous Function – A function whose domain has no gaps or disconnected values.  A sequence is a discrete function.

14-2 : Arithmetic, Geometric, and Other Sequences  Arithmetic Sequence – a sequence formed by adding the same number to each preceding term.  d is the common difference (the number added to all preceding terms)  Recursive Formula: a n =a n-1 +d  Explicit Formula: a n =a 1 +d(n-1)

14-2 : Arithmetic, Geometric, and Other Sequences  Arithmetic Sequences  3,5,7,9,…  Recursive formula: a 1 =3, a n =a n-1 +2  Explicit formula: a n =2n+1  5,2,-1,-4,…  Recursive formula:  Explicit formula:  What would the graph of the terms of an arithmetic sequence look like?

14-2 : Arithmetic, Geometric, and Other Sequences  Sum of an Arithmetic Series  S n =n / 2 (a 1 + a n )

14-2 : Arithmetic, Geometric, and Other Sequences  Geometric – a sequence formed by multiplying the same number to each preceding term.  r is the common ratio (the number multiplied by all preceding terms)  Recursive Formula: a n =r(a n-1 )  Explicit formula: a n =a 1 (r) n-1

14-2 : Arithmetic, Geometric, and Other Sequences  Geometric Sequences  3,6,12,24,…  Recursive formula: a 1 =3, a n =2a n-1  Explicit formula: a n =3(2) n-1  4,-8,16,-32,…  Recursive formula:  Explicit formula:  What would the graph of the terms of a geometric sequence look like?

14-2 : Arithmetic, Geometric, and Other Sequences  Sum of a Finite Geometric Series  S n = a 1 (1 – r n ) / (1 – r )  Sum of an Infinite Geometric Series  Abs value of r < 1  S n = a 1 / (1 – r )