For a given G.P., if T_5=405 and T_7=3645; then find T_4. - Vidyadip

Question

For a given $G.P.$, if $T_5=405$ and $T_7=3645$; then find $T_4$.

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Answer

$T _5=405, T_7=3645 ; T _4=?$
$T_n=a . r^{n-1}$
$\therefore T_5=a . r^4$
$\therefore 405=\text { a.r }{ }^4.........(1)$
$T_7= a . r ^6$
$\therefore 3645=\text { a.r }{ }^6.........(2)$
$T_4= a . r ^3..........(3)$
Taking the ratio of results $(2)$ and $(1),$
$\frac{3645}{405}=\frac{a r^6}{a r^4}$
$\therefore 9=r^2$
$\therefore r= \pm 3$
Putting $r= \pm 3$ in result $(1),$
$405= a ( \pm 3)^4$
$\therefore 405= a \times 81$
$\therefore a =\frac{405}{81}=5$
Putting $a=5$ and $r= \pm 3$ in result $(3),$
$\therefore T_4=5( \pm 27)$
$\therefore T_4= \pm 135$

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