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Graphing Sequences

Graphing Sequences. Sec. 9.4b. But first, we start with…. The fifth and ninth terms of an arithmetic sequence are –5 and –17, respectively. Find the first term and a recursive rule for the n -th term. The general explicit rule for an arithmetic sequence:. Plug in the given data:.

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Graphing Sequences

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  1. Graphing Sequences Sec. 9.4b

  2. But first, we start with… The fifth and ninth terms of an arithmetic sequence are –5 and –17, respectively. Find the first term and a recursive rule for the n-th term. The general explicit rule for an arithmetic sequence: Plug in the given data: A system to solve!!!

  3. But first, we start with… The fifth and ninth terms of an arithmetic sequence are –5 and –17, respectively. Find the first term and a recursive rule for the n-th term. First Term: Recursive Rule: Subtract the equations:

  4. Another similar problem: The third and sixth terms of a geometric sequence are –75 and –9375, respectively. Find the first term, common ratio, and an explicit rule for the n-th term. The general explicit rule for a geometric sequence: Plug in the given data: Another system to solve – Woo Hoo!!!

  5. Another similar problem: The third and sixth terms of a geometric sequence are –75 and –9375, respectively. Find the first term, common ratio, and an explicit rule for the n-th term. First Term: Divide the equations: Explicit Rule:

  6. Graphing Sequences We can represent sequences graphically in two ways: (a) As a scatter plot, and (b) Using the sequence graphing mode. Ex: Produce on a calculator a graph of the sequence in which METHOD 1 (Scatter Plot) The command seq(K, K, 1, 10)  L puts the first 10 natural numbers in list one. 1 2 The command L – 1  L puts the corresponding terms of the sequence in list two. 1 2 Now graph the scatter plot in window [–1, 15] by [–10, 100]!!!

  7. Graphing Sequences We can represent sequences graphically in two ways: (a) As a scatter plot, and (b) Using the sequence graphing mode. Ex: Produce on a calculator a graph of the sequence in which METHOD 2 (Sequence Mode) Put your calculator into Seq mode, then enter into the Y = list, with nMin = 1 and nMax = 10. Now graph the sequence in the same window!!!

  8. Graphing Sequences Using a graphing calculator, generate the specific terms of the following sequences: 1. (Explicit) for k = 1, 2, 3,… First Command: 0  K Second Command: K + 1  K:3K – 5 Then press ENTER repeatedly!!!

  9. Graphing Sequences Using a graphing calculator, generate the specific terms of the following sequences: for n = 2, 3, 4,… 2. (Recursive) First Command: –2 Second Command: ANS + 3 Then press ENTER repeatedly!!!

  10. A Famous Recursive Sequence The Fibonacci Sequence The Fibonacci sequence can be defined recursively by for all positive integers To get this sequence on your calculator: First Command: 0  A:1  B Second Command: A + B  C:A  B:C  A Then press ENTER repeatedly!!!

  11. Sums of Sequences

  12. Definition:Summation Notation In summation notation, the sum of the terms of the sequence is denoted which is read “the sum of from k = 1 to n.” The variable k is called the index of summation.

  13. Our First “Exploration” Determine the number represented by each of the following expressions. 3. 1. 4. 2. 5.

  14. Gauss’s Insight First, read the second-to-last paragraph on page 739… Your challenge is to find the sum of the natural numbers from 1 to 100 without a calculator. 1. Write the sum 2. Underneath this sum, write the sum 3. Add the numbers in each vertical column. You should get the same identical sum 100 times – what is it?

  15. Gauss’s Insight First, read the second-to-last paragraph on page 739… Your challenge is to find the sum of the natural numbers from 1 to 100 without a calculator. 4. What is the sum of the 100 identical numbers referred to in part 3? 5. Explain why half the answer in part 4 is the answer to the challenge. Can you find it without a calculator? The sum in part 4 involves two copies of the same progression, so it doubles the sum of the progression. The answer that Gauss gave was 5050.

  16. Let’s Prove the General Theorem Two ways to write an arithmetic sum: Sum these two expressions vertically:

  17. Let’s Prove the General Theorem for : Substitute

  18. Theorem:Sum of a Finite Arithmetic Sequence Let be a finite arithmetic sequence with common difference d. Then the sum of the terms of the sequence is

  19. Applying our New Theorem A corner section of a stadium has 8 seats along the front row. Each successive row has two more seats than the row preceding it. If the top row has 24 seats, how many seats are in the entire section? The number of seats in the rows forms an arithmetic sequence: Let’s solve for n:

  20. Applying our New Theorem A corner section of a stadium has 8 seats along the front row. Each successive row has two more seats than the row preceding it. If the top row has 24 seats, how many seats are in the entire section? Use the new formula: seats To find with your calculator: sum(seq(8 + (N – 1)2, N, 1, 9) = 144 seats

  21. Finding the Sum of a Geometric Sequence The general notation: Multiply both sides by r : Subtract the two summations:

  22. Finding the Sum of a Geometric Sequence Factor out common terms: Solve for the summation:

  23. Theorem:Sum of a Finite Geometric Sequence Let be a finite geometric sequence with common ratio r = 1. Then the sum of the terms in the sequence is

  24. Applying our Second New Theorem Find the sum of the geometric sequence given below. Identify terms: The new formula:

  25. Applying our Second New Theorem Find the sum of the geometric sequence given below. Support with a calculator: sum(seq(4(–1/3)^(N – 1), N, 1, 11) = 3.000016935

  26. Infinite Series

  27. First, let’s return to an example from last class… We found this sum: Now, we explore what happens when we change the “11” to INFINITY!!!

  28. First, let’s return to an example from last class… This is our first example of an infinite series, which is an expression where an infinite number of terms are added together………(duh?)

  29. Definition: Infinite Series An infinite series is an expression of the form Note: An infinite series is not a true sum…

  30. Definition: Infinite Series Sometimes a sequence of partial sums (all of which are true sums) approaches a finite limit S: In this case we say that the series converges to S, and we define S as the sum of the infinite series. In sigma notation, If the limit of the partial sums does not exist, then the series diverges and has no sum.

  31. Guided Practice For each of the following series, find the first five terms in the sequence of partial sums. Which of the series appear to converge? 1) 0.1 + 0.01 + 0.001 + 0.0001 + … First five partial sums: {0.1, 0.11, 0.111, 0.1111, 0.11111} These appear to approach a limit of 1/9  So the series converges to a sum of 1/9!!! 2) 10 + 20 + 30 + 40 + … First five partial sums: {10, 30, 60, 100, 150} These numbers approach no limit  The series diverges

  32. Guided Practice For each of the following series, find the first five terms in the sequence of partial sums. Which of the series appear to converge? 3) 1 – 1 + 1 – 1 + … First five partial sums: {1, 0, 1, 0, 1} These numbers oscillate and do not approach a limit  The series diverges NOTE: You cannot apply certain rules (such as the associate property of addition) to infinite series!!!

  33. Theorem:Sum of an Infinite Geometric Series A geometric series converges if and only if . If it does converge, the sum is .

  34. Guided Practice Determine whether the given series converges. If it converges, give the sum. 1)  The series converges First term: Sum:

  35. Guided Practice Determine whether the given series converges. If it converges, give the sum. 2)  The series converges First term: Sum:

  36. Guided Practice Determine whether the given series converges. If it converges, give the sum. 3)  The series diverges

  37. Guided Practice Determine whether the given series converges. If it converges, give the sum. 4)  The series converges First term: Sum:

  38. Guided Practice Express the given decimal in fraction form. We can write this number as a sum: This is a convergent geometric series with a = 0.234 and r = 0.001. The sum is:

  39. Guided Practice The table below shows the December balance in a simple interest savings account each year from 1996 to 2000. Year 1996 1997 1998 1999 2000 Balance $18,000 $20,016 $22,032 $24,048 $26,064 (a) The balances form an arithmetic sequence. What is d ? Find the difference between any two balances d = 2016 (b) Write a formula for the balance in the account n years after December 1996.

  40. Guided Practice The table below shows the December balance in a simple interest savings account each year from 1996 to 2000. Year 1996 1997 1998 1999 2000 Balance $18,000 $20,016 $22,032 $24,048 $26,064 (c) Find the sum of the December balances from 1996 to 2006, inclusive. Sum of the eleven terms of the arithmetic sequence:

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