8 1 Sequences. Defining a Sequence. A sequence {a n } is a list of numbers written in explicit order. e.g.

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1 8 1 Sequences Defining a Sequence A sequence {a n } is a list of numbers written in explicit order. e.g. If the domain is finite, then the sequence is a Ex.1 Explicit Sequence Find the first six terms and the 100 th term of the sequence {a n }, where This example is defined defined in terms of n. because it is

2 Recursive Sequences depend on what has gone on before. Ex. 2 Find the first three terms and the sixth term for the recursive sequence defined by the following conditions: b 1 =5 b n =2b n 1 3 for n 2 TRY Repeat example 2 with the following conditions: b 1 = 4 b 2 = 7 b n =3b n 2 + 2b n 1 for n 3

3 Arithmetic Sequences Each term in an arithmetic sequence can be obtained recursively from its preceding term by adding d. d is the Recursive definition: Explicit definition:

4 Ex. 3 Defining Arithmetic Sequences For the arithmetic sequence { 7, 3,1,5,9,...} find: (a) the common difference (b) a recursive rule for the nth term (c) an explicit rule for the nth term (d) the 42 nd term Ex.4 Repeat the above for the arithmetic sequence {ln2, ln6, ln18, ln54,...}

5 Ex. 5 Constructing a Sequence The third and sixth terms of an arithmetic sequence are 5 and 14, respectively. Find the common difference, first term, and an explicit rule for the nth term.

6 Geometric Sequences Each term in a geometric sequence can be obtained recursively from its preceding term by multiplying by r. r is the Recursive definition: Explicit definition:

7 Ex. 6 Defining Geometric Sequences For the geometric sequence {4, 12,36, 108,...} find: (a) the common ratio (b) a recursive rule for the nth term (c) an explicit rule for the nth term (d) the seventh term Ex. 7 Repeat example 6 for the geometric sequence {5 3,5 5,5 7,5 9,...}

8 Ex. 8 Constructing a Sequence The third and sixth terms of a geometric sequence are 20 and 160, respectively. Find the common ratio, first term, and an explicit rule for the nth term.

9 Limit of a Sequence Some sequences do not have a limit e.g. Some sequences tend towards a limit as n > e.g. Notation: L is the limit of the sequence. If L exists we say the sequence CONVERGES to L. Sequences that do not have limits DIVERGE. Properties of Limits If L and M are real numbers and and, then 1) Sum Rule: 2) Difference Rule: 3) Product Rule: 4) Quotient Rule: 5) Constant Multiple Rule:

10 Ex. 9 Finding the Limit of a Sequence Determine whether the sequence converges or diverges. If it converges, find its limit. (a) (b) Ex. 10 Determining Convergence or Divergence Determine whether the sequence with the given nth term converges or diverges. If it converges, find its limit. (a) n=1,2,3,... (b) b 1 =4, b n =b n 1 +2 n 2

11 Sandwich Theorem for Sequences If and there is an integer N for which a n b n c n for n>n, then. Ex. 11 Show that the sequence limit. converges, and find its Ex. 12 Determine if the sequence find its limit. converges, if it converges Factorials In your groups: Evaluate 4! Expand n! 3 different ways. What is? What is? Which one grows faster? What is? What is? Which one gets smaller more quickly?

12 Ex.13 Graphing using Parametric Mode Sometimes it helps to represent a geometric sequence graphically. It is a good idea to use parametric mode when graphing sequences. Draw a graph of the sequence {a n } with a n = Solution Let X 1 =T and Y 1 =( 1) T (T 1) 2 /T Graph in dot mode. T min =1 and T max =20 with T Step =1 X min =0 and X max =20 with X Scl =2 Y min = 20 and Y max =20 with Y Scl =1 Sketch what you see below:

13 Ex.14 Graphing using Sequence Graphing Mode Graph the sequence defined recursively by: b 1 =5 b n =b n 1 +3 n 2 Use Seq mode Use dot mode Replace b n by u(n) Parameters: Select nmin=1, U(n)=u(n 1)+3, U(nMin)=5 Window nmin=1, nmax=10, Plotstart=1, PlotStep=1 use the window [0,10] by [ 5,25] graph. Sketch what you see below: