Nth Term in a Sequence

Date: 01/11/2003 at 12:41:35
From: Xina
Subject: Sequence and series

Here is the sequence: 1, 2, 5, 14 ... 

Find the following 2 terms and a formula for the nth term. 

Are we expected to use trial and error to find the nth term, or are 
there any nice formulas to help us?

Date: 01/12/2003 at 00:15:37
From: Doctor Kastner
Subject: Re: Sequence and series

Hi Xina -

Sadly, there are no nice formulas that you can use to figure out these
sorts of problems. All is not lost, however; we just need to expand
our thinking a bit. Looking for a common difference is a good first
start, but what we really want to find is some type of pattern. Let's
think about the following sequence:

  1 3 6 10 15

If you look at the differences between terms, you'll see that it goes

  2 3 4 5

So while there isn't a common difference, there is a clear pattern to
the differences, and the next numbers would be 21 and 28. The same is
true for the sequence

  1 2 4 8 16 32

The differences are

  1 2 4 8 16

which again doesn't have a common term, but it is a nice pattern of
doubling. With this in mind, let's look back at your sequence. The
differences are

  1 3 9

which again suggests a pattern. Do you see the next number?

I hope this helps.  Write back if you're still stuck, or if you have
other questions.

- Doctor Kastner, The Math Forum
  http://mathforum.org/dr.math/