find the nth term?

Question

how can I find the nth term in sequence like this:
7, 25, 63, 129, 231, 377

Answer 1

I'm not entirely sure (yet), but notice what happens when you take the difference of each term and its successor:
18, 38, 66, 102, 146, ...
i.e., 7 + (18) = 25, 25 + (38) = 63, 63 + (66) = 129, ...

Now take the difference of each term and its successor in our new sequence:
20, 28, 36, 44, ...

Which looks to be linear; indeed, the formula would be f(n) = 8n + 12.

You might be able to use this to deduce the original sequence; I'm not quite sure of the method for doing this, but I feel it is reasonable to conjecture that the sequence is cubic in nature. If so, you can set up and solve this system of equations:
f(1) = 7 = a + b + c + d,
f(2) = 25 = 8a + 4b + 2c + d,
f(3) = 63 = 27a + 9b + 3c + d,
f(4) = 129 = 64a + 16b + 4c + d.

Good Luck!