Question 583415: Can you help me figure out what the formula is in this pattern: 8,24,72,216 ? Example of formula:3n-3 N=the position of the number
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! There are two special sequence patterns that are studies in algebra:
arithmetic sequences (called arithmetic progressions in other countries) and
geometric sequences (called geometric progressions in other countries).
In an arithmetic sequence each number (we call them terms) is the one before plus a constant that we call the common difference. To find out if a sequence is an arithmetic sequence, we calculate the differences between consecutive terms. In this case, we calculate:
 ,
 , and that is enough to realize that the differences (16 and 48) are not the same. There is no common difference. It is not an arithmetic sequence.
In a geometric sequence, each term is equal to the previous number times a constant that we call the common ratio. To find out if a sequence is a geometric sequence, we calculate the ratios of consecutive terms. In this case, we calculate:
 ,
 , and
 .
The ratio is always 3. There is a common ratio and it's 3. It is a geometric sequence. We call the nth term  , and we know that, calling the common ratio  ,
 or 
That's what we call a recursive formula.
We also know that, calling the first tern  ,
 , because
 ,
 ,
 , and so on.
We can do the formal proof by induction, if needed, but is a very useful formula to calculate the nth term.
In your case,
 ,  and
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