Picture of the exercise I'm having trouble with
So I'm supposed to do Question 14, and I'm having trouble with a). The theory that I've learned related to this is algebra: proof by mathematical induction. The question is asking me to find the nth term, so I'm not sure how I can use the proving with induction method. I'm somewhat certain that I could do b), but a) seems impossible for me at the moment. I've been trying to find some pattern in the series but so far I haven't cracked the code.
You can probably see that the numerator of our explicit equation is just going to be $n^2.$
As far as the denominator is concerned, note that the sequences $x_n=1,3,5,7,...$ and $y_n=3,5,7,9,...$ are just arithmetic sequences in which each subsequent term differs by $2.$ Recall that our formula for arithmetic sequences is $a_n=a_1+d(n-1)$ where $d$ is the common difference and $a_1$ is the first term in the sequence.
For $x_n$, $d=2$ and $x_1=1,$ so $x_n=1+2(n-1)=2n-1.$
For $y_n,$ $d=2$ and $y_1=3$, so $y_n=3+2(n-1)=2n+1.$
Putting this together, we get that $\frac{1^2}{1\cdot3}+\frac{2^2}{3\cdot5}+\frac{3^2}{5\cdot7}+\frac{4^2}{7\cdot9}+\cdots+\frac{n^2}{(2n-1)(2n+1)}$
You could have also just noted that $x_n=1,3,5,7,...$ and $y_n=3,5,7,9,...$ are just sequences of odd numbers, and odd numbers can always be expressed as $2k+1$ for some integer $k$, in which case it is fairly simple to derive the formulas we did above.