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Warm-up

Warm-up. Describe the pattern in each set of terms below, then find the next two terms of each set. 1. 2,4,6,8, … 2. 1,3,5,7, … 3. a, -b, c,- d, … 4. 1,4,9,16, … 5. , , , …. Warm-up. Describe the pattern in each set of terms below, then find the next two terms of each set.

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Warm-up

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  1. Warm-up Describe the pattern in each set of terms below, then find the next two terms of each set. 1. 2,4,6,8, … 2. 1,3,5,7, … 3. a, -b, c,- d, … 4. 1,4,9,16, … 5. , , , …

  2. Warm-up Describe the pattern in each set of terms below, then find the next two terms of each set. 1. 2,4,6,8, 10,12 Even numbers beginning at 2 2. 1,3,5,7, 9, 11 Odd numbers beginning at 1 3. a, -b, c, -d, e, -fAlternating signs of the alphabet from + to - 4. 1,4,9,16, 25, 36 Perfect square numbers starting at 1 5. , , , , , Rotating arrow clockwise starting from the east direction.

  3. Finite Versus Infinite Sequences • Describe the pattern of a sequence • Find a specified term of a sequence • The difference between a finite sequence and an infinite sequence • Use a graphing calculator to generate a sequence of numbers

  4. Let’s Review a Little • What is a function? • A relationship between two quantities in which each element of a set D, called the domain, is paired with exactly one element of a set, R, called the range. • What is the domain of a function? • The elements used as input values of the function (typically denoted as x-terms) • What is the range of a function? • The elements used as output values of the function (typically denoted as y-terms)

  5. Definitions • A sequence is a function whose domain is the set of positive integers. • Note: The function describes the pattern determined by the sequence • A sequence can be expressed as a list of numbers called terms of the sequence denoted as t1,t2,t3, … or by some rule or pattern such as tn = 2n.

  6. Ex1 Describe the pattern in the sequence 1, ½, , ¼, … a) Sketch the graph of this sequence. b) What do you notice? c) Write a rule for tn to find any term in the sequence above.

  7. Explicit Vs Recursive Definitions • A sequence expressed simply as a function of the position of the term is called an explicit definition. Ex tn = 2n explicitly defines the sequence of even numbers in the warm-up. Note: The explicit form of a sequence can be used to determine any term in the sequence.

  8. Ex2 Write the first five terms of the sequence & sketch its graph on the grid provided. What is t10? t20?

  9. Recursive Definitions • A sequence expressed by giving the value of a term in relation to the preceding term is called a recursive definition. • A recursive definition consists of two parts: • An initial condition that tells where the sequence starts. • A recursive equation (or formula) that tells how any term in the sequence is related to the preceding term. Ex t1 = 2; tn = 3tn-1 generates the sequence 2,6,18,…

  10. Ex4 The first term of a sequence is 2. Each successive term is one more than three times the previous term. • List the first five terms of the sequence. • Can you determine an explicit formula for this sequence?

  11. Finite Versus Infinite • A finite sequence contains a specific number of terms i.e. 1,2,3,4,5 • An infinite sequence continues without end i.e. 1,2,3,4,…

  12. Ex5 Determine the number of terms in each finite sequence. • 1,3,5,7,9 • 1,3,5,7,…15 • 2,4,8,16 • 2,4,8,….128

  13. Viewing Terms of a Sequence on the TI-83 • Check mode and range. • Clear y = • Make sure StatPlots are OFF. • Press 2nd button and then the STAT key to get to the LIST menu. • Go over to OPS and Select #5:SEQ • Type the sequence as follows: SEQ(explicit form, x, start position, stop position, increment amount) Ex SEQ(2x,x,5,10,2) gives 10, 14, 18

  14. Try this • Type in the expression given in Ex2 into your calculator. • Did you get the same terms we found earlier? Ex 3 Use your calculator to generate the tenth through the fifteenth terms of the sequence defined as tn = n + 1.

  15. Something to Think About • You aren’t feeling to well. In fact, you picked up a single virus Friday afternoon which has been doubling every day. How much of the virus do you expect in your system today? • Expressed as a sequence of numbers: 1,2,4,8, …

  16. Ex6 Determine an explicit definition for the following sequence: 1, -2, 3, -4, 5, -6, … • Notice signs of odd terms are positive • Notice signs of even terms are negative • Terms are simply the positions of each number with alternating signs • How can we generate alternating signs? tn = (-1)n+1n

  17. Tonight’s Assignment • Complete Handout on Finite Vs Infinite Sequences

  18. Exit Ticket • Share one thing you learned about sequences with a partner • Share one question you still have about sequences

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