# Unit 7 Section 4 : Formulae for General Terms

In the last section we learnt that we could use a formula which contains n to generate a sequence.
To do this we set n as the position in the sequence. For the first term we set n = 1 and for the second term we set n = 2 etc.

For the sequences 5n, 5n + 4 and 5n – 3 we get the following results:
 5n: 5, 10, 15, 20, 25, ... (the five times table) 5n + 4: 9, 14, 19, 24, 29, ... 5n – 3: 2, 7, 12, 17, 22, ...
We can see that in each case the sequence goes up by 5 each time, and it turns out that any sequence which goes up by 5 each time will contain 5n in the formula for the nth term of sequence.

 If the terms in a sequence go up by the same amount each time, then that amount will appear multiplied by n in the formula for the nth term of that sequence. e.g. If the terms in a sequence go up by 7 each time, then 7n will appear in the formula.

Example: Finding a general formula for a sequence

Imagine we want to find a formula for the nth term of this sequence: 7, 11, 15, 19, 23, ...

We can see that the terms in this sequence go up by 4 each time, so 4n must appear in the formula.
The sequence generated by the formula 4n is the four times table, but it isn't quite the sequence we want:
 4n sequence: 4,  8,  12, 16, 20, ... The original sequence we wanted: 7, 11, 15, 19, 23, ...
To get the terms in the original sequence, we need to add on 3 to each of the terms in the 4n sequence.

Hence, the general formula for the nth term of the original sequence is 4n + 3.

## Example Question

Work out the answer to each part of the question then click to see whether you are correct.

We want to find a general formula for the nth term of the following sequence:

5, 13, 21, 29, 37, ...

(a) What amount do the terms in the sequence go up by each time? (b) What will definitely appear in the formula for the nth term? (c) What sequence is generated by the general formula 8n? (d) What adjustment needs to be made to the 8n sequence to get the original sequence? (e) What is the general formula for the nth term of the original sequence? ## Exercises

Work out the answers to the questions below and fill in the boxes. Click on the button to find out whether you have answered correctly. If you are right then will appear and you should move on to the next question. If appears then your answer is wrong. Click on to clear your original answer and have another go. If you can't work out the right answer then click on to see the answer.

Question 1
For the sequence 2, 5, 8, 11, 14, . . .

(a) What is the difference between each term?  (b) What is the general formula for the nth term?  Question 2
For the sequence 6, 8, 10, 12, 14, . . .

(a) What is the difference between each term?  (b) What is the general formula for the nth term?  Question 3
For each sequence, write down the difference between each term and the general formula for the nth term.

 Difference between terms General formula for the nth term (a) 3, 5, 7, 9, 11, ...    (b) 5, 11, 17, 23, 29, ...    (c) 4, 7, 10, 13, 16, ...    (d) 7, 10, 13, 16, 19, ...    (e) 6, 10, 14, 18, 22, ...    Question 4

(a) What formula gives the sequence 4, 8, 12, 16, 20, ... ?  (b) What formula gives the sequence that is the multiples of 5?  Question 5

(a) What is the formula for the nth term of the sequence 7, 14, 21, 28, 35, ... ?  (b) What is the formula for the nth term of the sequence 8, 15, 22, 29, 36, ... ?  Question 6
Write down the formula for the nth term of each of the sequences below.
 (a) 3, 6, 9, 12, 15, ...  (b) 5, 12, 19, 26, 33, ...  (c) 21, 29, 37, 45, 53, ...  (d) 8, 11, 14, 17, 20, ...  (e) 3, 4, 5, 6, 7, ...  (f) 103, 106, 109, 112, 115, ...  You have now completed Unit 7 Section 4
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