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Math Help - proof of a constant sequence

  1. #1
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    proof of a constant sequence

    '''(an)=(p.n+q)/(r.n+s) is a constant sequence iff p/r=q/s''' how can ı prove this?
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  2. #2
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    Quote Originally Posted by sah_mat View Post
    '''(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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