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? 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 n + 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 terms  position  The difference between adjacent terms is difference 

? 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 n - 1 t n = 4n

? 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 n + 3 t n = 5n

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 , 1, 3, 5, 7,…..? 2 2 t n = 2n 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 from the common difference a. This is equivalent to asking yourself what you need to do to the common difference to get to the value of the first term.

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 = 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 = 453 Example Question 1

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 , 13, 18, 23, 28, t n = 5n + 3 t 100 = 5 x = 503 Question 2 1, 4, 7, 10, 13, t n = 3n - 2 t 100 = 3 x = 298 Question 3 2, 9, 16, 23, 30, t n = 7n - 5 t 100 = 7 x = 695 Question 4 9, 15, 21, 27, 33, t n = 6n + 3 t 100 = 6 x = 603 Question 5 -1, 4, 9, 14, 19, t n = 5n - 6 t 100 = 5 x = 494 Question 6 -3, 1, 5, 9, 13, t n = 4n - 7 t 100 = 4 x = 393 Question 7 6, 18, 30, 42, 54, t n = 12n - 6 t 100 = 12 x = 1194

Can you suggest why they are called linear sequences? t n = 2n + 1 t n = 3n - 4

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? S = 2D - 1S 10 = 2 x = 19 How many wooden braces (B) will there be, in the 20 th panel (P)? B = 3P + 1 B 20 = 3 x 20 +1=

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