Numeric number patterns G8 mathematics

Look at the following lists of numbers and see if you can predict each time what the next number will be:

(a) 2 4 6 8
(b) 121 132 143 154
(c) 3 12 27 48 
(d) 4 9 16 25
(e) 0 4 3 7 2 1 2 0 4 1
(f) 1 2 3 5 8 13 

Was there a list that you could not predict the next number for? Why not?
How did you decide in each case what the next number would be?

When a series of numbers follow each other according to a rule we talk of a number pattern or a sequence. And there are many examples in everyday life of such sequences. Being able to work out the rule and predict the sequence is very useful at times, for instance predicting how the population in an area will increase, how may more people are going to get COVID19, what the weather will be like and how much an investment may grow. Of course thes eexamples do not always follow and exact or regular rules, but we can still make intelligent guesses.

In the examples above the first sequence is easy to predict, because we see quickly that the numbers increase each time by 2. The second example may look more complicated, but it actually is very similar, the numbers increase each time by 11.

SOME TERMINOLOGY

The pattern is called a [sequence]
Each number in the sequence is called a term
We can say that in sequence (a) the second term is 4 or we can write it like this: T2  =  4

Look at the sequence: 11; 8; 5; 2;...

1. What are the next 3 terms?
2. What is the value of T4?
3. Which term is equal to -7?
4. What is the value of n if Tn = -1?

1. 11; 8; 5; 2; -1; -4; -7 (3 is subtracted each time)
2. T4 = 2 (2 is the fourth term in the sequence)
3. The seventh term T7 = -7
4. n=5 (T5 = -1)

Now look at the following rule: Tn = 3 X n + 1

1. What is the first term of the sequence?
2. What is the value of the 37th term?
3. What term has the value 34?
4. State the rule in words.

1. The first term is T1 so substitute 1 for n then Tn = 3 x n + 1 becomes T1 = 3 x 1 + 1 = 4
2. Substitute 37 for n then T37 = 3 x 37 +1 = 111 + 1 = 112
3. Tn = 3 x n + 1 = 34 So 3 X n = 34 -1 = 33 So n = 33 /3 = 11 Thus 34 is the 11th term
4. To obtain any term in the sequence multiply the term number by 3 and add 1.

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