Question

The sequence -1/5, 2/6, -3/7, 4/8How do I write a explicit equation for f(n) to represent the sequence.

Answer 1

So we must find an explicit equation for the sequence:

$-\frac{1}{5},\frac{2}{6},-\frac{3}{7},\frac{4}{8}$

If you look carefully, you'll see that not only each term is a fraction bu also the top number in the fraction of the n-th term is n. For example, in the first term we have a 1, in the second a 2, in the third a 3 and so on. Not only that, the bottom number in each fraction is the terms postion plus 4. The first term has a 1+4=5, the second a 2+4=6, the third a 3+4=7 and so on. The only thing left is that terms in odd positions are negative and terms in even positions are positive. A way to achieve this is by using the term:

$(-1)^n$

If we merge all of this into one equation we get:

$f(n)=(-1)^n\frac{n}{n+4}$

And that is the explicit equation we were looking for.