Question

see the attached file and answer it for me ​

Answer 1

MATE

YOUR SOLUTION IS IN ATTACHED PICTURE ☺️

Answer 2

Answer:

Let the given geometrical progression be:

a, ar, ar², ar³, .......

$(m + {n})^{th} term = a {r}^{m + n - 1} = p$

$(m - {n})^{th} term = a {r}^{m - n - 1} = q\:$

${r}^{2n} = \frac{p}{q}$

$r = ( \frac{p}{q} ) ^{ \frac{1}{2n} }$

Also,

${a}^{2} {r}^{2m - 2} = pq$

$a {r}^{m - 1} = \sqrt{pq} = {m}^{th} \: term \: of \: gp$

${n}^{th} \: term \: = a {r}^{n - 1} \\ \\ = \frac{p}{ {r}^{m + n - 1} } \times {r}^{n - 1} \\ \\ = \frac{p}{ {r}^{m} } \\ \\ = p \times {r}^{ - m} \\ \\ = p \times ( \frac{p}{q} ) ^{ \frac{ - m}{2n} } \\ \\ = p \times \frac{q}{p} ^{ \frac{m}{2n} }$

Hence,

It is Proved !