tn= 3n + 2 5, 8, 11, 14, 17,………………..? Linear Number Sequences/Patterns A linear number sequence is a sequence of numbers that has a constant difference between adjacent terms. Consider the first five terms of the number sequence shown: The difference between adjacent terms is 3. 3 3 3 3 difference  1st 2nd 3rd 4th 5th................................nth position  5, 8, 11, 14, 17,………………..? terms  + 2 We want to obtain a general rule that gives us the value of any term (nth) in the sequence as a function of the term’s position. n 3n 1 3 2 6 9 4 12 5 15 5 8 11 14 17 ? 3n + 2 Can you see how the numbers of this sequence are related to those in the 3 times table? Adjacent numbers in the 3 times table also differ by 3. The terms in this sequence are 2 bigger than the numbers in the 3 times table. tn= 3n + 2

5, 8, 11, 14, 17,…..? 1st 2nd 3rd 4th 5th.........nth 3 3 3 3 tn= 3n + 2 + 2 3, 7, 11, 15, 19,…..? tn= 4n - 1 - 1 4 4 4 4 n 4n 1 4 2 8 3 12 16 5 20 3 7 11 15 19 ? 4n - 1

5, 8, 11, 14, 17,…..? 1st 2nd 3rd 4th 5th.........nth 3 3 3 3 3, 7, 11, 15, 19,…..? 4 4 4 4 tn= 3n + 2 tn= 4n - 1 + 2 - 1 8, 13, 18, 23, 28,…..? tn= 5n + 3 5 5 5 5 + 3 n 5n 1 5 2 10 3 15 4 20 25 8 13 18 23 28 ? 5n + 3

tn= 3n + 2 tn= 4n - 1 tn= 5n + 3 tn= 2n - 3 5, 8, 11, 14, 17,…..? 5, 8, 11, 14, 17,…..? 1st 2nd 3rd 4th 5th.........nth 3 3 3 3 3, 7, 11, 15, 19,…..? 4 4 4 4 tn= 3n + 2 tn= 4n - 1 8, 13, 18, 23, 28,…..? 5 5 5 5 tn= 5n + 3 + 2 - 1 + 3 -1, 1, 3, 5, 7,…..? tn= 2n - 3 - 3 2 2 2 2 1. The common difference tells you the multiple of n required for the first part of the rule. 2. The second part of the rule is obtained by subtracting the first term and the common difference. 2a. This is equivalent to asking yourself what you need to do to the common difference to get to the value of the first term.

Example Question 1 For the number sequence below: (a) Find the “position to term” rule (b) Use your rule to find the 58th term (t58) Difference 7  7n 7  2  - 5 2, 9, 16, 23, 30,…… (a) tn= 7n - 5 (b) t58= 7 x 58 - 5 = 401 Example Question 2 For the number sequence below: (a) Find the “position to term” rule (b) Use your rule to find the 75th term (t75) 9, 15, 21, 27, 33,…… Difference 6  6n 6  9  + 3 (a) tn= 6n + 3 (b) t75= 6 x 75 + 3 = 453

For each of the number sequences below, find a rule for the nth term (tn) and work out the value of t100. Question 1 8, 13, 18, 23, 28, tn= 5n + 3 t100= 5 x 100 + 3 = 503 Question 2 1, 4, 7, 10, 13, tn= 3n - 2 t100= 3 x 100 - 2 = 298 Question 3 2, 9, 16, 23, 30, tn= 7n - 5 t100= 7 x 100 - 5 = 695 9, 15, 21, 27, 33, tn= 6n + 3 t100= 6 x 100 + 3 = 603 Question 4 Question 5 -1, 4, 9, 14, 19, tn= 5n - 6 t100= 5 x 100 - 6 = 494 -3, 1, 5, 9, 13, Question 6 tn= 4n - 7 t100= 4 x 100 - 7 = 393 Question 7 6, 18, 30, 42, 54, tn= 12n - 6 t100= 12 x 100 - 6 = 1194

Can you suggest why they are called linear sequences? 1 2 3 4 5 6 8 10 tn= 2n + 1 tn= 3n - 4

How many squares of chocolate (S) will the 10th diagram (D) contain? Number sequences can be used to solve problems involving patterns in diagrams. 1 2 3 4 How many squares of chocolate (S) will the 10th diagram (D) contain? 1 3 5 7 S = 2D - 1 S10 = 2 x 10 - 1 = 19 How many wooden braces (B) will there be, in the 20th panel (P)? 1 2 3 4 7 10 B = 3P + 1 B20 = 3 x 20 +1= 61

How many stone slabs (S) will the 15th diagram (D) contain? 2 3 How many stone slabs (S) will the 15th diagram (D) contain? 1 5 9 S = 4D - 3 S15 = 4 x 15 – 3 = 57 How many steel braces (B) will there be, in the 28th panel (P)? 1 2 3 6 11 16 B = 5P + 1 B28 = 5 x 28 + 1 = 141