Question

Prove that in a finite GP the product of the terms equidistant from the beginning and the end is always same and equal to the product of first and last term.

Answer 1

To prove finite GP the product of the terms equidistant from the beginning and the end is always same and equal to the product of first and last term:

Let us consider $a$ be the first term, $r$ be the common ratio and$1$ be the last term of a GP containing $n$ terms.

Then,

${\mathrm{p}}^{\mathrm{th}}$ term from the beginning $\times {\mathrm{p}}^{\mathrm{th}}$ term from the end$=p^{th}$ term from the beginning $\times (\mathrm{n}-\mathrm{p}+1)$the term from the beginning

$$\begin{gathered} =T_p\times T_{(n-p+1)} \\ =a{r}^{p-1}\times a{r}^{\left(n-p+1\right)-1} \\ =a{r}^{p-1}\times a{r}^{(n-p)} \\ =a\times a{r}^{(n-1)} \\ =T_1\times T_n \end{gathered}$$

Here $T_1=$first term and $T_2=$last term

Therefore, finite GP the product of the terms equidistant from the beginning and the end is always same and equal to the product of first and last term.

Hence proved.