I am currently trying to study difference equations for my first year undergrad Calculus course. I am struggling to understand how they work. I am currently trying this question:
$y_{n+1}=y_{n}*((n+1)/n)$
I tried using back-substitution ie $y_n=y_{n-2}*((n+1)/n)^2$, to get $y_n=((n+1)/n)^n*y_0$ but my textbook says the correct answer is $y_n=An$
Can you please outline how I go about solving difference equations and explain why An is the correct answer. Thanks.
We have $$\frac{y_{n+1}}{n+1}=\frac{y_n}{n}$$ Let $$S_n=\frac{y_n}{n}$$ Then $$S_{n+1}=S_n=A$$ for some constant $A$, so $$y_n=nS_n=An$$
Or, note that the product telescopes: $$y_n=y_{n-1}\frac{n}{n-1}=\frac{n}{n-1}\frac{n-1}{n-2}y_{n-2}=\frac{n}{n-1}\frac{n-1}{n-2}\frac{n-2}{n-3}\cdots=An$$