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Determine the explicit formula of the sequence:

$ 4/5, 6/7, 8/9, 10/11,... $

I am unsure how to make the explicit formula for this. Is there a formula to follow, or do i just have to look for a pattern?

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Generally, if you are given terms of a sequence you just have to look for a pattern. If the problem setter has been fair, once you find the correct (intended) pattern, it will seem obvious that you have the right one. Usually that means that the pattern has gone through enough repeats that it feels like nothing else as simple will work. T. Bongers has given a good hint. If you think this is sociology instead of mathematics, there is some truth there. – Ross Millikan Dec 16 '13 at 3:22
$$\frac{2n}{2n+1}\qquad,\qquad n\geqslant2$$ – Lucian Dec 16 '13 at 10:28
up vote 2 down vote accepted

Note that the denominators keep increasing by $2$ as do the numerators; perhaps it's a tad more suggestive to write

$$\frac 4 5 = 1 - \frac 1 5$$ $$\frac 6 7 = 1 - \frac 1 7$$ $$\frac 8 9 = 1 - \frac 1 9$$

and so on. I'll let you come up with the details of a specific formula.

As usual, there's the caveat that four terms don't uniquely define a sequence, so the formula really could be anything.

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