'''(an)=(p.n+q)/(r.n+s) is a constant sequence iff p/r=q/s''' how can ı prove this?
Hi
1) If it is a constant sequence then whatever n integer
$\displaystyle a_n = a_0$
$\displaystyle \frac{pn+q}{rn+s} = \frac{q}{s}$
$\displaystyle \frac{pn+q}{rn+s} - \frac{q}{s} = 0$
$\displaystyle \frac{psn+qs-qrn-qs}{s(rn+s)} = 0$
$\displaystyle \frac{n(ps-qr)}{s(rn+s)} = 0$
This equality being true whatever n
$\displaystyle ps-qr = 0$
2) If $\displaystyle \frac{p}{r} = \frac{q}{s}$ then
$\displaystyle a_n = \frac{pn+q}{rn+s} = \frac{pn+\frac{ps}{r}}{rn+s} = \frac{p(rn+s)}{r(rn+s)} = \frac{p}{r} = constant$