How do you Find the #n#-th term of the infinite sequence #1,-2/3,4/9,-8/27,…#?
When looking at sequences, your mind should always be in a pattern-recognizing mode. Sometimes sequences seem daunting, but when you look at them in pieces, they're usually easier to tackle.
With this sequence, we can see that for each term, the numerator doubles, and the denominator triples. Also, it's important to note that each successive term flips sign. (positive to negative to positive)
To tackle the sign flips, we know we'll have to put in our formula a term raising
Since each successive numerator is double the previous one, we can use
For the denominator, we'll use
Now, we simply piece them together: