Hi Shannon,

This is a very difficult question to address since there are so many different kinds of sequences. You can see what we have done in various cases if you go to the Quandaries and Queries page on Math Central and type nth term into the Quick Search window.

If the sequence is an arithmetic sequence then the nth term is

x_n = a + (n - 1) d

where a is the first term ( n = 1) and d is the common difference.

If the sequence is a geometric sequence then the nth term is

x_n = a r^n-1

where a is the first term and r is the common ratio.

If successive differences shows that the k^th difference is a constant then the nth term is

x_n = a₁ + a₂ n + a₃ n² + a₄ n³ + ··· + a_k+1 n^k

If you know k+1 terms you can use this expression to write k+1 linear equations that can be solved for the a-values. See for example our answer to Jenny's question which came a few years ago.

I hope this helps,
Penny