1. 40S Applied Math Mr. Knight – Killarney School Slide 1 Unit: Sequences Lesson: SEQ-L1 Sequences and Spreadsheets Sequences and Spreadsheets Learning Outcome B-4 SEQ-L1 Objectives: To identify and create sequences and their graphs using spreadsheets.

2. 40S Applied Math Mr. Knight – Killarney School Slide 2 Unit: Sequences Lesson: SEQ-L1 Sequences and Spreadsheets Monica was eligible to win a prize in a contest that required the contestants to describe patterns they observed around them. The mathematical part of the contest required the contestants to add two numbers to a sequence, and to explain the pattern. Monica was asked to write six answers for the five questions shown below: 5, 8, 11, 14, ___, ___ 3, 6, 12, 24, ___, ___ 1, 1, 2, 3, 5, 8, ___, ___ 3, 5, 7, 11, 13, 17, 19, 23, ___, ___ 1, 3, 7, ___, ___ In this lesson, you will learn to identify different types of sequences, study a number of definitions associated with sequences, and use spreadsheet formulas to write sequences and solve sequence problems similar to the ones shown above. Theory – Intro

3. 40S Applied Math Mr. Knight – Killarney School Slide 3 Unit: Sequences Lesson: SEQ-L1 Sequences and Spreadsheets Theory – Sequences A list of numbers that has a pattern is called a sequence. Once we know the pattern, we can predict any number in the sequence if we know the preceding number, or if we know which term it is in the sequence. Determine the patterns in the following sequences:

4. 40S Applied Math Mr. Knight – Killarney School Slide 4 Unit: Sequences Lesson: SEQ-L1 Sequences and Spreadsheets Once we know the pattern of a sequence, we can find any term of the sequence (use a SS). Answer the following questions: Theory – Find a Term

5. 40S Applied Math Mr. Knight – Killarney School Slide 5 Unit: Sequences Lesson: SEQ-L1 Sequences and Spreadsheets Using Spreadsheets As you could see on the previous page, writing a sequence to find a certain term can be a tedious task. Spreadsheets make this much easier and less time consuming. The example shows how a spreadsheet is used to find the 27th term of the sequence: 1, 1, 2, 3, 5, 8,... where you add two consecutive terms to find the next term. Step 1: Type the label 'Term' in cell A1, and the label 'Value' in cell B1. Step 2: Type the number '1' into cell A2 to represent the first term. Step 3: Type the formula: '=A2 + 1' into cell A3. This should represent the second term.

6. 40S Applied Math Mr. Knight – Killarney School Slide 6 Unit: Sequences Lesson: SEQ-L1 Sequences and Spreadsheets Using Spreadsheets cont’d Step 4: Use the 'Fill Down' feature to generate numbers in cells A4 to A28. (To do this, highlight cell A3, and click on the lower right corner of cell A3 and drag down to A28.) The numbers in cells A2 to A28 should be: 1, 2, 3, 4, · · ·, 27. Step 5: Type the number '1' into cell B2, and also into cell B3. Step 6: Type the formula: '=B2 + B3' into cell B4. Step 7: Fill cells B5 to B28 by clicking on cell B4 and filling down. The following slide shows part of the spreadsheet with numbers, as well as with formulas. The answer to the question "What is the 27th term" is '196,418'.

7. 40S Applied Math Mr. Knight – Killarney School Slide 7 Unit: Sequences Lesson: SEQ-L1 Sequences and Spreadsheets Using Spreadsheets cont’d

8. 40S Applied Math Mr. Knight – Killarney School Slide 8 Unit: Sequences Lesson: SEQ-L1 Sequences and Spreadsheets Theory – Recursive Spreadsheet Formulas The spreadsheet formulas used on the previous page were recursive formulas, because the value of each new term was based on the value of the previous term(s). For example, the formula '=A4+1' in cell A5 uses the value in cell A4 to generate the value in cell A5. Likewise, the formula '=B3+B4' in cell B5 uses the values in cells B3 and B4 to generate the value in B5. Sample problem: Given the sequence: 4, 7, 10, 13, 16, · · ·, and the spreadsheet shown below, write a suitable recursive formula in cell B3. Check your work by 'filling down' B3 to B7. Solution: A suitable recursion formula for cell B3 is =B2+3

9. 40S Applied Math Mr. Knight – Killarney School Slide 9 Unit: Sequences Lesson: SEQ-L1 Sequences and Spreadsheets Theory – Explicit Spreadsheet Formulas The sample problem from the previous page is repeated here. Given the sequence: 4, 7, 10, 13, 16, · · ·, and the spreadsheet shown, write a suitable formula in cell B3. Check your work by 'filling down' B3 to B7. This time we will write an explicit formula in cell B3. When writing an explicit formula, we use the position of the term in the sequence to write the formula. (We do not use the previous term in the sequence.) Note that this formula refers to A3 (the position in the sequence) and not to B2 (the previous term). Solution: A suitable explicit formula for cell B3 is =3*A3 + 1 One advantage of using an explicit formula is that you do not need to generate all the terms of a sequence to find the value of one particular term.

10. 40S Applied Math Mr. Knight – Killarney School Slide 10 Unit: Sequences Lesson: SEQ-L1 Sequences and Spreadsheets Practice – Recursive and Explicit Formulas In this question, you are asked to create a sequence by writing recursive and explicit spreadsheet formulas. The sequence is: 3, 8, 13, 18, 23, 28,... to 10 terms. Step 1: Type the labels 'Term' in cell A1, 'Value-R' in cell B1, and 'Value-E' in cell C1. Step 2: Type '1' in cell A2, and '3' in cell B2. Step 3: Type '=A2+1' in cell A3, and fill down A3 to A11 (for 10 terms). Step 4: Write a suitable recursive formula in cell B3, and fill down to B11. Step 5: Write a suitable explicit formula in cell C2, and fill down to C11.

11. 40S Applied Math Mr. Knight – Killarney School Slide 11 Unit: Sequences Lesson: SEQ-L1 Sequences and Spreadsheets Practice Solution - Recursive and Explicit Formulas

12. 40S Applied Math Mr. Knight – Killarney School Slide 12 Unit: Sequences Lesson: SEQ-L1 Sequences and Spreadsheets A arithmetic sequence is a sequence where any term after the first term can be found by adding a constant to the preceding term. The difference between successive terms will always be the same. For example, the sequence 4, 7, 10, 13, · · · is an arithmetic sequence because each term (after the first) may be found by adding '3' to the preceding term. In this case, the difference between successive terms is: difference = 13 - 10 = 10 - 7 = 3 Theory – Arithmetic Sequences

13. 40S Applied Math Mr. Knight – Killarney School Slide 13 Unit: Sequences Lesson: SEQ-L1 Sequences and Spreadsheets A geometric sequence is a sequence where any term after the first term can be found by multiplying a constant with the preceding term. The ratio between successive terms will always be the same. For example, the sequence 3, 6, 12, 24, · · · is a geometric sequence because each term (after the first) can be found by multiplying the preceding term by '2'. In this case, the ratio of successive terms is: Note that the definitions for arithmetic and geometric sequences given on this page are both recursive definitions. Theory – Geometric Sequences

14. 40S Applied Math Mr. Knight – Killarney School Slide 14 Unit: Sequences Lesson: SEQ-L1 Sequences and Spreadsheets In a diverging geometric sequence, the values of successive terms increase or decrease in size at an increasing rate. For example, in the geometric sequence 1, 3, 9, 27, 81, · · ·, the values of successive terms increase rapidly. In the geometric sequence –2, -4, -8, -16, · · ·, the values of successive terms decrease rapidly. A geometric sequence will be divergent if the multiplying number is greater than ‘1’. In a converging geometric sequence, the values of the sequence approach ‘zero’ as the number of terms increases. For example, in the geometric sequence 32, 16, 8, 4, · · ·, the value of each term decreases (i.e. approaches zero) as the number of terms increases. A geometric sequence will be converging if the multiplying number is between ‘0’ and ‘1’. Sample question: Write the sequence 24, 12, 6, 3,... using a spreadsheet. Is the sequence converging or diverging? What is the approximate value of the 18 th term? Theory – Diverging and Converging Sequences

15. 40S Applied Math Mr. Knight – Killarney School Slide 15 Unit: Sequences Lesson: SEQ-L1 Sequences and Spreadsheets Theory – Diverging and Converging Sequences

16. 40S Applied Math Mr. Knight – Killarney School Slide 16 Unit: Sequences Lesson: SEQ-L1 Sequences and Spreadsheets Create a spreadsheet with the sequence shown. The sequence should include 10 terms. Use the graphing function of a spreadsheet to draw a graph of the sequence, where 'Term' is the independent variable (x), and 'Value' is the dependent variable (y). Can you predict the shape of the graph? Theory – The Graph of an Arithmetic Sequence

17. 40S Applied Math Mr. Knight – Killarney School Slide 17 Unit: Sequences Lesson: SEQ-L1 Sequences and Spreadsheets Theory – The Graph of an Arithmetic Sequence

18. 40S Applied Math Mr. Knight – Killarney School Slide 18 Unit: Sequences Lesson: SEQ-L1 Sequences and Spreadsheets Create a spreadsheet with the following sequence. The sequence should have 10 terms. (Multiply each term by 1.5 to get the next term.) Use the graphing function of a spreadsheet to draw a graph of the sequence, where 'Term' is the independent variable, and 'Value' is the dependent. Can you predict the shape of the graph? Theory – The Graph of an Geometric Sequence

19. 40S Applied Math Mr. Knight – Killarney School Slide 19 Unit: Sequences Lesson: SEQ-L1 Sequences and Spreadsheets Theory – The Graph of an Geometric Sequence