Finding explicit formula of a series

Question

I'm trying to find the explicit formula of a series: $a_1=\frac{1}{2}, a_{n+1}=\frac{(n+1)^2}{n(n+2)}a_n$

I calculated, that $,a_2=\frac{9}{6},a_3=\frac{3}{5},a_4=\frac{5}{8}$.

However, I fail to find any logic in the series. Could you help me?

Thanks

Answer 1

We have $$\frac{(n+2)a_{n+1}}{n+1}=\frac{(n+1)a_n}{n}.$$ Let $b_n=\frac{(n+1)a_n}{n}.$

Thus, $$b_{n+1}=b_n,$$ which gives $$b_n=b_1=\frac{2}{2}=1.$$

Id est, $$a_n=\frac{n}{n+1}.$$

Answer 2

If it is a series like $\sum_n \frac{(n + 1)^2}{n (n + 1)}$, use partial fractions, just like when computing integrals.

If it is a sequence, there are several strategies that work. Here the simplest is to write out (don't simplify yet!) a few terms, and guess at the general form:

$\begin{align*} a_1 &= \frac{1}{2} \\ a_2 &= \frac{2^2}{1 \cdot 3} \cdot \frac{1}{2} \\ a_3 &= \frac{3^2}{2 \cdot 4} \cdot \frac{2^2}{1 \cdot 3} \cdot \frac{1}{2} \end{align*}$

You see what the numerators look like: Product of successive squares, i.e., the square of a factorial. Denominators are a bit more complex, the $n (n + 2)$ gives products of $1 \cdot 3 \cdot \dotsm$ and products $2 \cdot 4 \cdot \dotsm$. They interleave, do a few more, it seems it will be another factorial. If so, the fraction should simplify nicely, giving a simple formula for $a_n$.

OK, guessing's over. Prove your formula using induction. If it doesn't work, back to refining the guess.

Optionally, for bragging in front of freshmen: Write the original recursion, then "From here it is obvious that", your above formula, and the induction proof "for the incredulous".