I've been having some trouble identifying the formulas for some of the more difficult sequences in my textbook.
Most sequences I've seen come in fractions, which is nice to do. I always try and identify some kind of pattern between each numerator and each denominator, and find a relation between them and $n$. It's worked well, but some of these more difficult ones are getting the better of me. For instance:
$1: \{-3, 2, -\frac43, \frac89, -\frac{16}{27},...\}$
Normally, when I see $-,+,-,+,..$ I assume a negative base, with $n$ somewhere in the exponent, generally because that allows it to switch between positive and negative. Here, you see that happening, which is a good start. I also notice how the numerator from $n=2$ continues to double, and how the denominator continues to triple from $n=3$, but the $-3$ and the $-2$ completely throw that off.
Another thing to note, is that there is a possiblility of this formula being in the format $\{a_{n+1}\}_{n=1}^{infinity}$. I usually see this when the first few terms are whole numbers that are previously defined, and then suddenly throws itself into fractions, so that's also a possibility.
Other than those few clues, I'm completely thrown off, especially by the jump between $n=2$ and $n=3$.
Help with this question is appreciated, but I would also appreciate any tips on ways to go about identifying patterns in sequences. I know there isn't a formula way, but if there any tactics that are considered useful and worth noting, I would definitely love to hear them.
Much thanks