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Problem:

Problem:. Add the first 100 counting numbers together. 1 + 2 + …+ 99 + 100. We shall see if we can find a fast way of doing this problem. A pattern of numbers in a particular order is called a number sequence , and the individual numbers in the sequence are called terms.

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  1. Problem: Add the first 100 counting numbers together. 1 + 2 + …+ 99 + 100 We shall see if we can find a fast way of doing this problem. Designed by David Jay Hebert, PhD

  2. A pattern of numbers in a particular order is called a number sequence, and the individual numbers in the sequence are called terms. Designed by David Jay Hebert, PhD

  3. Each number or term in the sequence is associated with a position, which is also a number. Designed by David Jay Hebert, PhD

  4. Examples of Sequences 1,2,3,4,5,…. 1,1,2,3,5,8,… 1,2,4,8,16,32,… 2,1,7,3,9,3,… The dots at the end of the sequence indicate that the sequence continues without end. Note not all sequences have a pattern. Designed by David Jay Hebert, PhD

  5. Arithmetic Sequences The sequence 2, 5, 8, 11, 14,… Has first differences of 3 all the time, this makes this sequence arithmetic. If a sequence is arithmetic the first differences must be the same. Designed by David Jay Hebert, PhD

  6. First Differences First differences are the differences found by subtracting two consecutive numbers in a sequence. Ex. 2, 5, 8, 11, 14,…. Designed by David Jay Hebert, PhD

  7. First Differences First differences are the differences found by subtracting two consecutive numbers in a sequence. Ex. 2, 5, 8, 11, 14,…. 5 – 2 = 3 3 Designed by David Jay Hebert, PhD

  8. First Differences First differences are the differences found by subtracting two consecutive numbers in a sequence. Ex. 2, 5, 8, 11, 14,…. 8 – 5 = 3 3 3 Designed by David Jay Hebert, PhD

  9. First Differences First differences are the differences found by subtracting two consecutive numbers in a sequence. Ex. 2, 5, 8, 11, 14,…. 11 - 8 = 3 3 3 3 Designed by David Jay Hebert, PhD

  10. First Differences First differences are the differences found by subtracting two consecutive numbers in a sequence. Ex. 2, 5, 8, 11, 14,…. Since all the first differences are the same this must be an arithmetic sequence 3 3 3 3 Designed by David Jay Hebert, PhD

  11. Position Numbers Each term in the sequence is paired with a position number 2, 5, 8, 11, 14,… Designed by David Jay Hebert, PhD

  12. Arithmetic Sequences 2, 5, 8, 11, 14,… (recall the first difference is 3) 5 = 2 + 3 Designed by David Jay Hebert, PhD

  13. Arithmetic Sequences 2, 5, 8, 11, 14,… (recall the first difference is 3) 5 = 2 + 3 8 = 5 + 3 = 2 + 3 + 3 (the underlined part is the previous term) Designed by David Jay Hebert, PhD

  14. Arithmetic Sequences 2, 5, 8, 11, 14,… (recall the first difference is 3) 5 = 2 + 3 8 = 5 + 3 = 2 + 3 + 3 11 = 8 + 3 = 2 + 3 + 3 + 3 (the underlined part is the previous term) Designed by David Jay Hebert, PhD

  15. Arithmetic Sequences 2, 5, 8, 11, 14,… (recall the first difference is 3) 5 = 2 + 3 8 = 5 + 3 = 2 + 3 + 3 11 = 8 + 3 = 2 + 3 + 3 + 3 14 = 11 + 3 = 2 + 3 + 3 + 3 + 3 (the underlined part is the previous term) Designed by David Jay Hebert, PhD

  16. Arithmetic Sequence: Predictability The question is how many times does one move from the first term in a sequence. Ex. 2, 5, 8, 11, 14,…. Take one step from 2 to get to 5 Take two steps from 2 to get to 8 Take three steps from 2 to get to 11 Designed by David Jay Hebert, PhD

  17. Arithmetic Sequences • Have a constant first difference • Are predictable Designed by David Jay Hebert, PhD

  18. Predictability If the sequence in question is arithmetic the position numbers will begin with a 0th position. i.e., 2, 5, 8, 11, 14, … Designed by David Jay Hebert, PhD

  19. Predictability If the sequence in question is arithmetic the position numbers will begin with a 0th position. i.e., 2, 5, 8, 11, 14, … 2 is in the 0th position. Zero 3’s were added to 2 to get to 2. But 2 is the 1st term in the sequence. Designed by David Jay Hebert, PhD

  20. Predictability If the sequence in question is arithmetic the position numbers will begin with a 0th position. i.e., 2, 5, 8, 11, 14, … 5 is in the 1st position. One 3 was added to 2 to get to 5. But 5 is the 2nd term in the sequence. Designed by David Jay Hebert, PhD

  21. Predictability If the sequence in question is arithmetic the position numbers will begin with a 0th position. i.e., 2, 5, 8, 11, 14, … 8 is in the 2nd position. Two 3’s were added to 2 to get to 8. But 8 is the 3rd term in the sequence. Designed by David Jay Hebert, PhD

  22. Predictability If the sequence in question is arithmetic the position numbers will begin with a 0th position. i.e., 2, 5, 8, 11, 14, … 11 is in the 3rdposition. Three 3’s were added to 2 to get to 11. But 11is the 4th term in the sequence. Designed by David Jay Hebert, PhD

  23. Predictability If we wanted to know the term in the 22nd position, we would add 22 threes to 2 to get the result, i.e., 22(3) +2 = 66 + 2 = 68. 68 is in the 22nd position, but it is the 23rd term in the sequence. Designed by David Jay Hebert, PhD

  24. Predictability Mathematics is the study of patterns, and therefore we must look for a pattern. The position number is the same as the number of first differences that must be added to the first term. This leads me to believe that the following formula might work. Term = (Position Number)(Step Size) + Initial Step Designed by David Jay Hebert, PhD

  25. Arithmetic Sequences Start with a sequence -2, 3, 8, 13, 18, 23,… Find the step size (first difference). 3 – (-2) = 5 Find the initial step (term in 0th position) -2 Designed by David Jay Hebert, PhD

  26. Arithmetic Sequences Term = (position)(step size) + initial step From above Term= (position)(5) + -2 This is a relationship between the position of a term and the term in a position. Designed by David Jay Hebert, PhD

  27. Arithmetic Sequences Term = (position)(step size) + initial step From above Term= (position)(5) + -2 This is a relationship between the position of a term and the term in a position. What is the 45th term? Designed by David Jay Hebert, PhD

  28. Arithmetic Sequence Term = (position)(5) – 2 Term = (45)(5) – 2 Term = 210 – 2 Term = 208 By knowing the position we are able to determine the term, what about the other way around. Designed by David Jay Hebert, PhD

  29. Arithmetic Sequence What if we know the term but not the position, can the position be determined? Suppose that 183 is a number in the sequence –2, 3, 8, 13, 18,… Designed by David Jay Hebert, PhD

  30. Arithmetic Sequence The relationship Term = (Position)(Step Size) + Initial Step Replace with known values 183 = (Position)(5) – 2 37 = Position Designed by David Jay Hebert, PhD

  31. New Problem: Find the sum of the sequence: -2, 3, 8, 13, …,603, 608, 613 Designed by David Jay Hebert, PhD

  32. New Problem: S = -2 + 3 + 8 + … + 603 + 608 + 613 Is the same as S = 613 + 608 + 603 + … + 8 + 3 + (-2) Designed by David Jay Hebert, PhD

  33. New Problem: Notice the following S = -2 + 3 + 8 + … + 603 + 608 + 613 S = 613 + 608 + 603 + … + 8 + 3 + (-2) Add down in columns defined by the plus signs. Designed by David Jay Hebert, PhD

  34. New Problem: Notice the following S = -2 + 3 + 8 + … + 603 + 608 + 613 S = 613 + 608 + 603 + … + 8 + 3 + (-2) 2S = 611 + 611+ 611 + … + 611 + 611 + 611 If we knew the number of terms in the original sequence we could answer the question 611+ 611 + … + 611 + 611. Designed by David Jay Hebert, PhD

  35. New Problem: Notice the following S = -2 + 3 + 8 + … + 603 + 608 + 613 S = 613 + 608 + 603 + … + 8 + 3 + (-2) 2S = 611 + 611+ 611 + … + 611 + 611 + 611 The sequence –2, 3, 8, 13,… Is arithmetic therefore we have predictability. Designed by David Jay Hebert, PhD

  36. New Problem: Notice the following S = -2 + 3 + 8 + … + 603 + 608 + 613 S = 613 + 608 + 603 + … + 8 + 3 + (-2) 2S = 611 + 611+ 611 + … + 611 + 611 + 611 613 = (Position)(5) – 2 123 = Position, which means there are 124 terms in the sequence. Designed by David Jay Hebert, PhD

  37. New Problem: Notice the following S = -2 + 3 + 8 + … + 603 + 608 + 613 S = 613 + 608 + 603 + … + 8 + 3 + (-2) 2S = 611 + 611+ 611 + … + 611 + 611 + 611 There are 124 - 611’s in the sequence Designed by David Jay Hebert, PhD

  38. New Problem: Notice the following S = -2 + 3 + 8 + … + 603 + 608 + 613 S = 613 + 608 + 603 + … + 8 + 3 + (-2) 2S = 611 + 611+ 611 + … + 611 + 611 + 611 2S = (611)(124) Designed by David Jay Hebert, PhD

  39. New Problem: Notice the following S = -2 + 3 + 8 + … + 603 + 608 + 613 S = 613 + 608 + 603 + … + 8 + 3 + (-2) 2S = 611 + 611+ 611 + … + 611 + 611 + 611 2S = (611)(124) 2S = 75764 Designed by David Jay Hebert, PhD

  40. New Problem: Notice the following S = -2 + 3 + 8 + … + 603 + 608 + 613 S = 613 + 608 + 603 + … + 8 + 3 + (-2) 2S = 611 + 611+ 611 + … + 611 + 611 + 611 2S = (611)(124) 2S = 75764 S = 37882 Designed by David Jay Hebert, PhD

  41. New Problem: Let us find the sum of the sequence 5, 9, 13, … , 365, 369, 373 First we must find out how many terms are in the sequence. In order to do this we must determine the step size and initial step of the sequence. Designed by David Jay Hebert, PhD

  42. New Problem: Let us find the sum of the sequence 5, 9, 13, … , 365, 369, 373 Step Size = 4 Initial Step = 5 Hence Term = 4(Position) + 5 Designed by David Jay Hebert, PhD

  43. New Problem: Let us find the sum of the sequence 5, 9, 13, … , 365, 369, 373 Term = 4(Position) + 5 373 = 4(Position) + 5 92 = Position We will need this information later. Designed by David Jay Hebert, PhD

  44. Step 1: Write down the sum you wish to determine S = 5 + 9 + 13 + … + 365 + 369 + 373 Designed by David Jay Hebert, PhD

  45. Step 2: reverse the order of the sum and write it under the first sum. S = 5 + 9 + 13 + … + 365 + 369 + 373 S = 373 + 369 + 365 + …+ 13 + 9 + 5 Designed by David Jay Hebert, PhD

  46. Step 3: add down in columns determined by the plus signs in the problem. S = 5 + 9 + 13 + … + 365 + 369 + 373 S = 373 + 369 + 365 + …+ 13 + 9 + 5 2S = 378 + 378 + 378+ … + 378 + 378 + 378 Notice that all the columns have the same sum, if only we knew how many they were! Designed by David Jay Hebert, PhD

  47. Step 4: recall we had predictability with arithmetic sequences and we already determined that 373 was in position 92 of the original sequence. Therefore 93 terms in the sequence. S = 5 + 9 + 13 + … + 365 + 369 + 373 S = 373 + 369 + 365 + …+ 13 + 9 + 5 2S = 378 + 378 + 378+ …+ 378 + 378 + 378 Designed by David Jay Hebert, PhD

  48. Step 5: replace the long sum with the associated multiplication problem. S = 5 + 9 + 13 + … + 365 + 369 + 373 S = 373 + 369 + 365 + …+ 13 + 9 + 5 2S = 378 + 378 + 378+ …+ 378 + 378 + 378 2S = 378(93) Designed by David Jay Hebert, PhD

  49. Step 6: solve for S and be done with the problem. S = 5 + 9 + 13 + … + 365 + 369 + 373 S = 373 + 369 + 365 + …+ 13 + 9 + 5 2S = 378 + 378 + 378+ …+ 378 + 378 + 378 2S = 378(93) 2S = 35154 S = 17577 Designed by David Jay Hebert, PhD

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