Let
![a_{0}](/media/m/5/5/d/55d5676473434b5cc90ef28741998c0b.png)
,
![a_{1}](/media/m/0/6/5/0653090dabb5d1972cd7a7dfcd31abc1.png)
,
![a_{2}](/media/m/5/5/6/5565dac5c7f1dadb0e60c273c1d11c80.png)
,
![...](/media/m/8/1/8/818bbd7bfc42992ed89357baed116aeb.png)
be a sequence of reals such that
![a_{0} = - 1](/media/m/c/2/a/c2a3aa2a6bcc970723c6d2187285fcca.png)
and
![a_{n} + \frac {a_{n - 1}}{2} + \frac {a_{n - 2}}{3} + ... + \frac {a_{1}}{n} + \frac {a_{0}}{n + 1} = 0](/media/m/6/a/1/6a131dd06addba57d889e8d452594e90.png)
for all
![n\geq 1](/media/m/6/0/b/60b196d3e8aa7fce08d72404eea76d0e.png)
.
Show that
![a_{n} > 0](/media/m/1/4/1/14180513db2f467e47f2b15e0730a114.png)
for all
![n\geq 1](/media/m/6/0/b/60b196d3e8aa7fce08d72404eea76d0e.png)
.
%V0
Let $a_{0}$, $a_{1}$, $a_{2}$, $...$ be a sequence of reals such that $a_{0} = - 1$ and
$a_{n} + \frac {a_{n - 1}}{2} + \frac {a_{n - 2}}{3} + ... + \frac {a_{1}}{n} + \frac {a_{0}}{n + 1} = 0$ for all $n\geq 1$.
Show that $a_{n} > 0$ for all $n\geq 1$.