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Q.
In a sequence if the sum of the first $n$ terms is given by $S_n=2^{ np }-1$ where ' $p$ ' is fixed real number. The nature of the sequence, is
Sequences and Series
Solution:
$ t _{ n +1}= S _{ n +1}- S _{ n }=\left(2^{( n +1) p }-1\right)-\left(2^{ np }-1\right)=2^{ np } \cdot 2^{ p }-2^{ np }=2^{ np }\left(2^{ p }-1\right)$
$||| l y\,\,\,\,\, t _{ n }= S _{ n }- S _{ n -1}=\left(2^{ np }-1\right)-\left(2^{( n -1) p }-1\right)=2^{( n -1) p }\left(2^{ p }-1\right) $
$\therefore \frac{ t _{ n +1}}{ t _{ n }}=\frac{2^{ np }}{2^{( n -1) p }}=2^{ p }=\text { constant }$
Hence sequence is G.P.