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  • If the p\textsuperscript{th} and q\textsuperscript{th} terms of a G.P. are q and p respectively, show that its (p plus q)\textsuperscript{th} term is

If the  p\textsuperscript{th} and \: \: q\textsuperscript{th} terms of a G.P. are q and p respectively, show that its (p + q)\textsuperscript{th} term is \left(\frac{q^{p}}{p^{4}}\right)^{\frac{1}{p-8}}

Answers (1)

The nth term is given by t_{n}=ar^{n-1}

The pth term is given by t_{p}=ar^{p-1}

q=ar^{p-1}

\frac{q}{r^{p}}=\frac{a}{r}

t_{q}=ar^{q-1}

p=ar^{q-1}

\\ \frac{p}{r^{q}}=\frac{a}{r}\\\\ \frac{p}{r^{q}}=\frac{q}{r^{p}}\\\\ r^{p-q}=\frac{q}{p}\\

r= \left( \frac{q}{p} \right) ^{\frac{1}{p-q}}

t_{p+q}=a \left( r \right) ^{p+q-1}

t_{p+q}= \left( ar^{p-1} \right) r^{q} \\\\ t_{p+q}=qr^{q} \\\\ t_{p+q}=q \left( \left( \frac{q}{p} \right) ^{\frac{1}{p-q}} \right) ^{q} \\\\

t_{p+q}=q \left( \frac{q^{\frac{q}{p-q}}}{p^{\frac{q}{p-q}}} \right) = \left( \frac{q^{1+\frac{q}{p-q}}}{p^{\frac{q}{p-q}}} \right) = \left( \frac{q^{\frac{p}{p-q}}}{p^{\frac{q}{p-q}}} \right) = \left( \frac{q^{p}}{p^{q}} \right) ^{\frac{1}{p-q}} \\\\

 

 

 

 

 

 

Posted by

infoexpert21

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