proof of a constant sequence

**sah_mat** · December 14th 2008, 08:25 AM

'''(an)=(p.n+q)/(r.n+s) is a constant sequence iff p/r=q/s''' how can ı prove this?

**running-gag** · December 14th 2008, 09:12 AM

Originally Posted by sah_mat

'''(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
$a_n = a_0$

$\frac{pn+q}{rn+s} = \frac{q}{s}$

$\frac{pn+q}{rn+s} - \frac{q}{s} = 0$

$\frac{psn+qs-qrn-qs}{s(rn+s)} = 0$

$\frac{n(ps-qr)}{s(rn+s)} = 0$

This equality being true whatever n
$ps-qr = 0$

2) If $\frac{p}{r} = \frac{q}{s}$ then

$a_n = \frac{pn+q}{rn+s} = \frac{pn+\frac{ps}{r}}{rn+s} = \frac{p(rn+s)}{r(rn+s)} = \frac{p}{r} = constant$

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