Finding the Rule For a Times Something Add Something Sequence

A 'times something then add something sequence' takes a term, multiplies it by something then adds something to find the next term.

If the rule is 'multiply a term by 3 then add 2' and the first term is 4,

the second term is 3*4+2=14

the third term is 3*14+2=44

the fourth term is 3*44+2=134

Given a sequence, to find the rule we must find what to multiply by and what to add.

We can find what to multiply by by dividing terms by the previous terms – the second term by the first term, the third term by the second term, the fourth term by the second term and so on. If these tend to a specific number, then this is the number we must multiply by. We can then work out what to add.

Example: Find the rule for the sequence 5, 19, 75, 299, 1195,...

We are obviously getting closer and closer to 4, so our rule starts 'multiply by 4...'.

To work out what to add (or take away) note that 4*5=20 but the second term is 19, so we must subtract one. The rule is 'multiply a term by four then subtract one to get the next term'.

It make no difference to reverse the order of operation. An 'add something times something' sequence is in fact also a 'times something add something sequence'. Suppose the rull is 'add two then times three', so that if a term is x, the next term will be 3(x+2)=3x+6, so we could rephrase the rule as 'times three then add six', the sequences are equivalent.