f(n)

bigman · Poincare Conjecture Offline Joined: 18 Aug 2008 Posts: 183

Let $f$ be the function defined on all positive integers by
$f(1) = 1$ and $f(1)+f(2)+...+f(n)=n^2f(n)$ for all $n>=2$ . Find and prove
an explicit formula for $f(n)$ .

**Posted:** Wed Nov 04, 2009 1:15 pm

tonypr

Solution

**Posted:** Wed Nov 04, 2009 2:39 pm

_________________
Math in trivial problem solving

Tomekk

Hello,

We have $f(1) + f(2) + ... + f(n) = n^{2}f(n)$ and $f(1) = 1$ .

From first formula we can get :

$f(1) + f(2) + ... + f(n - 1) = (n^{2} - 1)f(n)$

and

$f(1) + f(2) + ... + f(n - 1) = (n - 1)^{2}f(n - 1)$

From there we get:

$(n^{2} - 1)f(n) = (n - 1)^{2}f(n - 1)$

And

$f(n) = \frac {(n - 1)}{(n + 1)}f(n - 1)$

$f(n) = \frac {(n - 1)}{(n + 1)}\frac {(n - 2)}{n}\frac {(n - 3)}{(n - 1)}.....\frac {1}{3}$

Finally we get $f(n) = \frac {2}{n(n + 1)}$ and we are done . Smile

**Posted:** Wed Nov 04, 2009 2:45 pm