The problem is to find out the next term of the series {$-3$, $2$, $-4/3$, $8/9$, $-16/27$}.
I have tried various things but haven't been able to come up with an answer. Is there a method to solve questions like these? Or is it just intuition? Please help.
The question actually required me to find the general term.
Best Answer
The next term problem is generally not a math problem, there is an infinity of possibilities, unless you spot a pattern. Possibly, starting from $k=0$, you have: $$s_k = -(-2)^k.3^{1-k}\,,$$ or $$s_k = -3 \left(\frac{-2}{3}\right)^k\,.$$ Works for the five first terms.
In your case, it is easy to spot powers of $2$ and powers of $3$, and the alternating sign. Powers suggest computing ratios of terms. The alternating sign suggests it is not too complicated, because $-1,1,-1,-1,\ldots$ is not.
Simple techniques are computing differences or ratios, or differences of differences, etc. And recognizing standard series. But this falls short fast with $5$ terms.
But you never know what the person who asks you the question has in mind. What would one do with $8,12,15,20,23,28,32,35,38,43,45,50,56$?
This is the cumulated number of letters from words of the lyrics of Shine on you crazy diamond, by the Pink Floyd, which I am currently listening to. So the next is $59$, because "now" follows "Remember when you were young, you shone like the sun. Shine on you crazy diamond".