Finding the Formula for a Sequence

[Ayesh]

I just wanted to know if anybody had a trick to find the formula for the general term of a sequence.

I can find some without troubles. But other like 1,0,1,0,1,0... I don't know how to find them.

Thank you!

 

[LCKurtz]

I just wanted to know if anybody had a trick to find the formula for the general term of a sequence.

I can find some without troubles. But other like 1,0,1,0,1,0... I don't know how to find them.

Thank you!

Of course you know that technically, the next term could be anything and the first few terms of a sequence never determines a pattern. So these questions always involve a subjective judgement or guess what the author intends. In this case one suspects the described sequence is to consist of alternating 1's and 0's. It is more a matter of experience than anything to come up with a formula. This might suggest sin(n pi/2) for n = 1,2,..., which gives:

1 0 -1 0 1...

Not quite it. Put absolute values on it: |sin (n pi/2)|:

1 0 1 0 ... works.

Another thing you might think of is using (-1)ⁿ, which alternates the sign:

-1 1 -1 1 -1 1...

Try adding 1 to it: 1 + (-1)ⁿ: 0 2 0 2 0 2...

Close. Divide it by 2: (1 + (-1)ⁿ)/2: 0 2 0 2 0 2 if you start with n = 1. So start with n = 0.

Each problem presents its own challenges.