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Q.
In a G.P. if $( m + n )^ {th }$ term is $p$ and $( m - n )^ {th }$ term is $q$, then $m ^ {th }$ term is:
Sequences and Series
Solution:
For a G. P , $a _{ m + n }= p$ and $a _{ m - n }= q$,
We know that $a _{ n }= AR ^{ n -1}$ (in G.P.)
where A= first term and $R=$ ratio
$\therefore a _{ m + n }= p$
$ \Rightarrow AR ^{ m + n -1}= p\, \dots(i)$
and $a_{m-n}=q$
$ \Rightarrow A R^{m-n-1}=q\, \dots(ii)$
On multiply equations (i) and (ii), we have
$\left( AR ^{ m + n -1}\right) \cdot\left( AR ^{ m - n -1}\right)= pq $
$\Rightarrow A ^{2} \cdot R ^{2( m -1)}= pq$
$\Rightarrow \left( AR ^{ m -1}\right)^{2}= pq$
$ \Rightarrow AR ^{ m -1}=\sqrt{ pq }$
$\Rightarrow a _{ m }=\sqrt{ pq }$