Patterns – Learning Outcomes Express patterns in words and numbers. Use patterns to continue sequences. Find terms in arithmetic sequences. Express patterns in algebraic form. Use tables and graphs to represent patterns. Find the sum to 𝑛 of an arithmetic series. Recognise whether a sequence is arithmetic, geometric, quadratic, or none of these.

Explain Patterns in Words and Numbers Consider the sequence: 3, 9, 15, 21, 27, … Imagine each number in a box in sequence: 3 9 15 21 27 Box #1 Box #2 Box #3 Box #4 Box #5 Draw out the next three boxes. What number is in box #10? What box contains the number 75? What number is in box n?

Explain Patterns in Words and Numbers 3 9 15 21 27 Box #1 Box #2 Box #3 Box #4 Box #5 The number in box #1 is called the start term, a. The difference between boxes is called the common difference, d. The number in box #n is called the nth term, Tn. (e.g. box #5 contains the 5th term, T5, box #25 contains the 25th term, T25). Only arithmetic patterns have a common difference!

Explain Patterns in Words and Numbers Write the first five terms of the following sequences: start term = 2, common difference = 5 start term = 5, common difference = 2 start term = 18, common difference = -2 Describe in words how to find the nth term of each of these sequences. Write an expression in terms of n to describe the nth term of these sequences.

Explain Patterns in Words and Numbers Write the first four terms of each of the following sequences: 𝑇 𝑛 =5𝑛+1 𝑇 𝑛 =2𝑛+4 𝑇 𝑛 =9 4−2𝑛 Write down the start term and the common difference for each of the sequences above.

Find Terms in Arithmetic Sequences 2004 OL P1 Q5a The first term of an arithmetic sequence is 40 and the common difference is –5. Write down the first five terms of the sequence. 2005 OL P1 Q5a The first term of an arithmetic sequence is 9 and the second term is 13. Find the common difference. Find the third term. 2006 OL P1 Q5a The first term of an arithmetic sequence is 17 and the common difference is -8. Find, in terms of 𝑛, an expression for 𝑇 𝑛 , the 𝑛th term.

Find Terms in an Arithmetic Sequence 2006 OL P1 Q5c The first three terms of an arithmetic sequence are ℎ+3, 5ℎ − 2, 6ℎ − 13, where ℎ is a real number. Find the value of ℎ. Hence, write down the value of each of the first three terms. Find the value of the eleventh term.

Use Tables to Represent Arithmetic Patterns Disc-shaped tiles are placed to form a pattern as shown: Draw the next two stages of the pattern. Draw a table showing how many tiles are in each stage of the pattern. Write down the general term, 𝑇 𝑛 , for the number of tiles in stage 𝑛 of the pattern. 𝑇 𝑛 =𝑎+ 𝑛−1 𝑑

Use Tables to Represent Arithmetic Patterns 2017 FL Q10 The following diagram shows an arrangement of tables and chairs in a sequence of patterns. Draw the 4th pattern in the sequence. Create a table to show the number of chairs in each of the first 6 patterns. Use your data from part (b) to graph the relationship between the number of tables and the number of chairs. How many chairs are there in the 10th pattern?

Use Tables to Represent Arithmetic Patterns There are exactly 54 chairs in one of the patterns. How many tables are in that pattern? How many chairs, in total, are there in the first 7 patterns? Write a formula (in words) that shows the relationship between the number of chairs and the number of tables in any given pattern.

Use Tables to Represent Arithmetic Patterns 2013 FL P1 Q10 Jim builds a fence by using three horizontal rails between each two vertical posts. Jim draws the diagrams below and begins to draw up a table to show the number of rails he will need depending on how many posts he uses.

Use Tables to Represent Arithmetic Patterns Complete the table above. Jim thinks that to find the number of rails needed he should subtract 1 from the number of posts used and multiply the answer by 3. Write an algebraic expression to represent Jim’s rule, using 𝑥 to represent the number of posts and 𝑦 to represent the number of rails. Test your expression in (b) above using the numbers in one row of the table. Jim uses 60 posts for his fence. Find the number of rails he needs.

Use Tables to Represent Arithmetic Patterns Ann thinks that an alternative rule to find the number of rails is to multiply the number of posts by 3 and then subtract 3 from the answer. Write an algebraic expression to represent Ann’s rule, using 𝑥 to represent the number of posts and 𝑦 to represent the number of rails. Use Ann’s rule to find how many rails are needed if 10 posts are used. Use Ann’s rule to find how many posts are used if 228 rails are needed.

Find 𝑎 and 𝑑 When Given Terms In an arithmetic sequence, 𝑇 2 =5 and 𝑇 7 =20. Find the start term and the common difference. For an arithmetic sequence, 𝑇 3 =11 and 𝑇 6 =26. Find 𝑎 and 𝑑. If 𝑇 2 =20 and 𝑇 6 =0 are terms in an arithmetic sequence, find 𝑎, 𝑑, and 𝑇 8 . If 𝑥, 𝑥+2, and 3𝑥+2 are consecutive terms in an arithmetic sequence, find the common difference. Three consecutive terms in an arithmetic sequence are 3𝑥+25, 2𝑥+10, and 3𝑥−55. Find 𝑑.

Find the Sum of an Arithmetic Series An arithmetic series is the sum of the terms of an arithmetic sequence. e.g. Given the sequence 4, 7, 10, 13, 16, 19… 𝑆 1 =4 𝑆 2 =4+7=11 𝑆 3 =4+7+10=21 𝑆 4 =4+7+10+13=34 𝑆 5 =4+7+10+13+16=50 𝑆 𝑛 =4+7+10+13+16+…

Find the Sum of an Arithmetic Series e.g. Given the sequence 1, 3, 5, 7, 9, … Find 𝑆 1 Find 𝑆 3 Find 𝑆 5 e.g. Given the sequence 11, 7, 3, -1, -5, … Find 𝑆 2 Find 𝑆 𝐾 , where 𝑆 𝐾 is the first negative term in the series.

Find the Sum of an Arithmetic Series The general formula for a sum of an arithmetic series is: 𝑆 𝑛 = 𝑛 2 [2𝑎+ 𝑛−1 𝑑] , where 𝑎 is the start term, 𝑑 is the common difference, and 𝑛 is the term number. e.g. The first term of an arithmetic sequence is 40 and the common difference is -5. Find 𝑆 𝑛 Hence, find 𝑆 10 .

Find the Sum of an Arithmetic Series 𝑆 𝑛 = 𝑛 2 [2𝑎+ 𝑛−1 𝑑] Find 𝑆 3 , 𝑆 5 , and 𝑆 𝑛 for each of the following sequences: 5, 7, 9, 11, … 2, 5, 8, 11, … 1, 6, 11, 16, … 4, 8, 12, 16, … 18, 13, 8, 3, … 10, 8, 6, 4, … 20, 10, 0, -10, …

Find the Sum of an Arithmetic Series 2003 OL P1 Q5c The fifth term of an arithmetic series is 21 and the tenth term is 11. Find the first term and the common difference. Find the sum of the first twenty terms. For what value of 𝑛>0 is the sum of the first 𝑛 terms equal to 0? 2004 OL P1 Q5b The 𝑛th term of an arithmetic sequence is given by 𝑇 𝑛 = 1+5𝑛. Find the first term and the common difference. Find the value of 𝑛 for which 𝑇 𝑛 =156. Find 𝑆 12 , the sum of the first 12 terms.

Find the Sum of an Arithmetic Series 2005 OL P1 Q5b The sum of the first 𝑛 terms of an arithmetic series is given by 𝑆 𝑛 = 𝑛 2 +𝑛. Find 𝑎, the first term. Find 𝑆 2 , the sum of the first two terms. Find 𝑑, the common difference. Write down the first five terms of the series.

Find the Sum of an Arithmetic Series Disc-shaped tiles are placed to form a pattern as shown: Find 𝑆 𝑛 Find 𝑆 10 If each stage of the pattern is made using tiles and earlier patterns are not broken up to make later ones, what is the last stage of the pattern that can be made with 230 tiles?

Use Tables to Represent Quadratic Patterns Linear patterns have a common difference, d. Quadratic patterns have a common second difference. e.g. 3, 6, 11, 18, 27… Term 3 6 11 18 27 1st Diff +3 +5 +7 +9 2nd Diff +2

Use Tables to Represent Quadratic Patterns Confirm that the following sequences are quadratic and write down the next two terms: 1, 4, 9, 16, 25… 3, 6, 11, 18, 27… 0, 3, 8, 15, 24… 2, 8, 18, 32, 50… -3, 8, 23, 42, 65… 9, 28, 57, 96, 145… 3, 12, 27, 48, 75… 16, 7, 2, 1, 4…

Use Tables to Represent Quadratic Patterns The term rules for quadratic sequences are in the form 𝑎 𝑛 2 +𝑏𝑛+𝑐. Write down the first four terms of each of the following sequences: 𝑇 𝑛 = 𝑛 2 +2𝑛+1 𝑇 𝑛 = 𝑛 2 −2𝑛+1 𝑇 𝑛 = 𝑛 2 +3𝑛−6 𝑇 𝑛 =2 𝑛 2 +5𝑛−10 𝑇 𝑛 =3 𝑛 2 −6𝑛+4 𝑇 𝑛 =7 𝑛 2 −7𝑛+7

Use Tables to Represent Quadratic Patterns When given terms in a quadratic sequence, we can generate the term rule 𝑎 𝑛 2 +𝑏𝑛+𝑐 using three facts: 𝑎= 𝑠𝑒𝑐𝑜𝑛𝑑 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 2 By letting 𝑛=0, we get 𝑐= 𝑇 0 . Although this does not normally exist, we can pretend it does for this purpose. 𝑏 can be found by substitution.

Use Tables to Represent Quadratic Patterns e.g. 3, 6, 11, 18, 27 (from pg 23) Term 3 6 11 18 27 1st Diff +3 +5 +7 +9 2nd Diff +2 𝑎= 𝑠𝑒𝑐𝑜𝑛𝑑 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 2 = 2 2 =1 Working backwards, 𝑇 0 =3−1=2⇒𝑐=2 e.g. 𝑇 1 = 1 1 2 +𝑏 1 +2=3 ⇒1+𝑏+2=3 ⇒𝑏=0 ⇒ 𝑇 𝑛 = 𝑛 2 +2 Recall that in general, 𝑇 𝑛 =𝑎 𝑛 2 +𝑏𝑛+𝑐 for quadratic sequences

Use Tables to Represent Quadratic Patterns Determine the term rule for each of the following quadratic sequences (you can look up the answers from pg 23 to speed this up): 1, 4, 9, 16, 25… 3, 6, 11, 18, 27… 0, 3, 8, 15, 24… 2, 8, 18, 32, 50… -3, 8, 23, 42, 65… 9, 28, 57, 96, 145… 3, 12, 27, 48, 75… 16, 7, 2, 1, 4…

Use Tables to Represent Geometric Patterns Geometric sequences do not have a common difference, but instead have a common factor. e.g. 2, 6, 18, 54, 162… Term 2 6 18 54 162 1st Diff +4 +12 +36 +108 2nd Diff +8 +24 +72 Neither the first nor second differences are common, so try factors: Term 2 6 18 54 162 Factor ×3

Use Tables to Represent Geometric Patterns Confirm that the following sequences are geometric and write down the next two terms: 6, 18, 54, 162… 6, 12, 24, 48… 10, 20, 40, 80… 12, 36, 108, 324… -4, -8, -16, -32… -9, -27, -81, -243…

Use Tables to Represent Geometric Patterns Write out the first four terms of each of the following geometric sequences: 𝑇 𝑛 = 2 𝑛 𝑇 𝑛 = 3 𝑛 𝑇 𝑛 =4 3 𝑛 𝑇 𝑛 =5 2 𝑛 𝑇 𝑛 =10 2 𝑛 𝑇 𝑛 =7 3 𝑛

Use Tables to Represent Geometric Patterns 2011 OL P1 Q5a The first term of a geometric sequence is 5 and the common ratio is 2. Find the first four terms of the sequence. 2009 OL P1 Q5a The first term of a geometric sequence is 2 and the common ratio is 3. Find the second term of the sequence. 2008 OL P1 Q5b The nth term of a geometric sequence is 𝑇 𝑛 = 3 𝑛 27 Find the first term. Find the common ratio.