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3. 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: 5, 8, 11, 14, 17,………………..? 1 st 2 nd 3 rd 4 th 5 th................................n th We want to obtain a general rule that gives us the value of any term (n th ) in the sequence as a function of the term’s position. Can you see how the numbers of this sequence are related to those in the 3 times table? n3n3n 13 26 39 412 515 3n + 2 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. t n = 3n + 2 + 2 terms  position  The difference between adjacent terms is 3. 3 3 difference 

4. 3 7 11 15 19 ? 5, 8, 11, 14, 17,…..? 1 st 2 nd 3 rd 4 th 5 th.........n th 3 3 3, 7, 11, 15, 19,…..? 4 4 t n = 3n + 2 n4n4n 14 28 312 416 520 4n - 1 t n = 4n - 1 + 2 - 1

5. 8 13 18 23 28 ? 5, 8, 11, 14, 17,…..? 1 st 2 nd 3 rd 4 th 5 th.........n th 3 3 3, 7, 11, 15, 19,…..? 4 4 t n = 3n + 2 t n = 4n - 1 8, 13, 18, 23, 28,…..? 5 5 n5n5n 15 210 315 420 525 5n + 3 t n = 5n + 3 + 2 - 1 + 3

6. 5, 8, 11, 14, 17,…..? 1 st 2 nd 3 rd 4 th 5 th.........n th 3 3 3, 7, 11, 15, 19,…..? 4 4 t n = 3n + 2 t n = 4n - 1 8, 13, 18, 23, 28,…..? 5 5 t n = 5n + 3 -1, 1, 3, 5, 7,…..? 2 2 t n = 2n - 3 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. + 2 - 1 + 3 - 3 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.

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

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

9. Can you suggest why they are called linear sequences? 1 2 34 0 5 2 4 6 8 10 t n = 2n + 1 t n = 3n - 4

10. Number sequences can be used to solve problems involving patterns in diagrams. How many squares of chocolate (S) will the 10 th diagram (D) contain? 1 2 3 4 1357 S = 2D - 1S 10 = 2 x 10 - 1 = 19 How many wooden braces (B) will there be, in the 20 th panel (P)? 1 2 3 B = 3P + 1 B 20 = 3 x 20 +1= 61 4 710

11. How many stone slabs (S) will the 15 th diagram (D) contain? 1 2 3 159 S = 4D - 3S 15 = 4 x 15 – 3 = 57 How many steel braces (B) will there be, in the 28 th panel (P)? 1 2 3 611 16 B = 5P + 1B 28 = 5 x 28 + 1 = 141