Question

If the $p^{th}$ term of a G.P. is $\alpha$ and qth term is $\beta$ then nth term is :

• A. ${\left(\frac{{\alpha }^{n-q}}{{\beta }^{n-p}}\right)}^{\frac{1}{p-q}}$
• B. ${\left(\frac{{\alpha }^{n-q}}{{\beta }^{n-p}}\right)}^{\frac{1}{q-p}}$
• C. ${\left(\frac{{\alpha }^{n-p}}{{\beta }^{n-q}}\right)}^{\frac{1}{q+p}}$
• D. ${\left(\frac{{\alpha }^{n-p}}{{\beta }^{n-q}}\right)}^{\frac{1}{q-p}}$

Answer 1

The correct option is A ${\left(\frac{{\alpha }^{n-q}}{{\beta }^{n-p}}\right)}^{\frac{1}{p-q}}$

$p^{th}$ term of G.P. $=a{r}^{p-1}=\alpha$

$q^{th}$ term of G.P. $=a{r}^{q-1}=\beta$

$r^{p-q}=\frac{\alpha }{\beta }$

$r={\frac{\alpha }{\beta }}^{\frac{1}{p-q}}$

$a{r}^{p-1}=a({\frac{\alpha }{\beta }}^{\frac{1}{p-b}}{)}^{p-1}=\alpha$

$a={\alpha }^{\frac{1-q}{p-q}}{\beta }^{\frac{p-1}{p-q}}$

$n^{th}$ term of G.P. $=a{r}^{n-1}={\frac{{\alpha }^{n-q}}{{\beta }^{n-p}}}^{\frac{1}{p-q}}$