The nth term of a series is given by `t_(n)=(n^(5)+n^(3))/(n^(4)+n^(2)+1)` and if sum of its n terms can be expressed as `S_(n)=a_(n)^(2)+a+(1)/(b_(n)^(2)+b)` where `a_(n)` and `b_(n)` are the nth terms of some arithmetic progressions and a, b are some constants, prove that `(b_(n))/(a_(n))` is a costant.

Explanation

To solve the problem step by step, we will start with the given nth term of the series and then derive the required expressions to prove that \(\frac{b_n}{a_n}\) is a constant.

### Step 1: Simplify the nth term \( t_n \)

The nth term is given by:

\[
t_n = \frac{n^5 + n^3}{n^4 + n^2 + 1}
...