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Q.
The first and eight terms of a G.P. are $x^{-4}$ and $x^{52}$ respectively. If the second term is $x^t$, then t is equal to:
Sequences and Series
Solution:
Let a be the first term and r be the common ratio so, general term of G.P is $T_n = ar^{n-1}$
As given,
$T_{1} =x^{-4} =a\, and\, T_{8 } =ar^{7} =x^{52} \therefore ar^{7} =x^{52}$
$ \Rightarrow x^{-4} r^{7} = x^{52} \Rightarrow r^{7} = x^{56}$
$ \Rightarrow r^{7} = \left(x^{8}\right)^{7} \Rightarrow r = x^{8} $
$\therefore T_{2} =ar^{1} = x^{-4} . x^{8} $
$T_{2} = x^{4} $
But $T_{2} = x^t \Rightarrow x^{t} = x^{4} \Rightarrow t = 4 $