-4
$\begingroup$

I have this:

f(1) = 1
f(n) = f(n-1)*(n/(n+2))

I have to convert it to it's explicit form, but I have absolutely no idea

Edit: the first elements are

1       0.3333333333333333 
2       0.16666666666666666 
3       0.1
4       0.06666666666666667
5       0.047619047619047616
6       0.03571428571428571
7       0.027777777777777776 
8       0.02222222222222222 
9       0.01818181818181818
$\endgroup$
5
  • 2
    $\begingroup$ Did you compute $f(2)$, $f(3)$ and $f(4)$? $\endgroup$ Commented Sep 6, 2022 at 22:20
  • $\begingroup$ I did, they are 1/3,1/6,1/10,1/15 $\endgroup$
    – szg12345
    Commented Sep 6, 2022 at 22:22
  • 2
    $\begingroup$ I don't think so. $\endgroup$ Commented Sep 6, 2022 at 22:25
  • 1
    $\begingroup$ So does $f(1) = 1$ or $f(1) = 0.3333333333333333$? $\endgroup$
    – peterwhy
    Commented Sep 6, 2022 at 22:48
  • 1
    $\begingroup$ @AnneBauval. Bonjour, voisine ! $\endgroup$ Commented Sep 7, 2022 at 12:31

1 Answer 1

-1
$\begingroup$

$f(2)=f(1)\cdot \frac{2}{4}=\frac{2}{4}=\frac{1}{2}$

$f(3)=f(2)\cdot \frac{3}{5}=\frac{2}{4}\cdot \frac{3}{5}=\frac{3}{10}$

$f(4)=f(3)\cdot \frac{4}{6}=\frac{2}{4}\cdot \frac{3}{5}\cdot \frac{4}{6}$

$f(n)=\frac{\prod_{k=2}^{n}k}{\prod_{k=4}^{n+2}k}=\frac{6}{(n+1)(n+2)}$.

$\endgroup$

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