Question

Write the explicit equation for f(n) to represent the sequence -1/5, 2/6, -3/7, 4/8

Answer 1

`f(n)=(-1)^n\cdot(n)/(n+4)`

Step-by-step 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:

`f(n)=(n)/(n+4)`

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:

`f(n)=(-1)^n\cdot(n)/(n+4)`

Let's see if it works fine by obtaining the first 4 terms:

`f(1)=(-1)^1\cdot(1)/(1+4)=-(1)/(5)`
`f(2)=(-1)^2\cdot(2)/(2+4)=(2)/(6)`
`f(3)=(-1)^3\cdot(3)/(3+4)=-(3)/(7)`
`f(4)=(-1)^4\cdot(4)/(4+4)=(4)/(8)`

The equation gives the same 4 terms given.