Respuesta :
ANSWER
[tex]f(n)=(-1)^n\cdot\frac{n}{n+4}[/tex]EXPLANATION
Note that the denominators and the numerators (without the negative sign) are all continuous. The first term has a denominator 5, which is n + 4 and this is true for all of the fraction's denominators.
Then the numerators are the number of term itself: n.
For now we have:
[tex]f(n)=\frac{n}{n+4}[/tex]But this gives all positive numbers. To get the odd numbers negative and the even numbers positive, we use (-1)^n. This, when n is odd gives -1 as a result and when n is even, gives 1 as a result.
Therefore the equation is:
[tex]f(n)=(-1)^n\cdot\frac{n}{n+4}[/tex]Let's see if it works fine by obtaining the first 4 terms:
[tex]f(1)=(-1)^1\cdot\frac{1}{1+4}=-\frac{1}{5}[/tex][tex]f(2)=(-1)^2\cdot\frac{2}{2+4}=\frac{2}{6}[/tex][tex]f(3)=(-1)^3\cdot\frac{3}{3+4}=-\frac{3}{7}[/tex][tex]f(4)=(-1)^4\cdot\frac{4}{4+4}=\frac{4}{8}[/tex]The equation gives the same 4 terms given.
