nth Term Rule (Deductive Rule)

Is there a certain way of finding the nth term rule of a sequence?

I really struggle finding it for sequences - it seems to just be a trial and error attempt of playing around with the first term until achieving the second, then testing that out on the next term (and so on).

As an example with this question:

Write down a deductive rule in the form U_r = ... for the general term of the sequence 1, 5, 9, 13, 17, ...

Most of the examples you'll be given will have a relatively straightforward form that you will have encountered in class.

In this case you want to be looking at the difference between each term...

I'm self studying so haven't seen anything in class

The sequences have been straightforward but the only method I know is going from term 1 to term 2, finding a rule, then trying it out on the next term, and so on. Not very efficient for a sequence with 100's of terms in though lol

But they're not going to give you a sequence with "100s of terms in" are they??

The point is that they'll give you something that has to be deducible from 4 or 5 terms otherwise it wouldn't be manageable and no-one would get it in the exam

I see, so it is just a case of trial and error?

A constant difference between terms is linear (arithmetic sequence), a constant second difference is quadratic sequence. If you keep differencing and the sequence still grows at the same rate, it's exponential/geometric (look at the ratio between terms, rather than the difference) ... You should have met this before?
As davros says, with 4 or 5 terms, it can't be too complex a sequence.

I see, so it is just a case of trial and error?

A constant difference between terms is linear (arithmetic sequence), a constant second difference is quadratic sequence. If you keep differencing and the sequence still grows at the same rate, it's exponential/geometric (look at the ratio between terms, rather than the difference) ... You should have met this before?
As davros says, with 4 or 5 terms, it can't be too complex a sequence.

PRSOM

As explained above, it's not really "trial and error" - it's learning to recognize different types of sequences that are covered by your syllabus and the methods used to identify them.

Be a bit wary of TSR - people tend to throw all sorts of sequences on here: some are standard exam questions, some have typos, some are competition standard, and some are made up by teachers to keep pupils occupied and don't follow any simple mathematical rule