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8. Sequences and Series
medium
$1, 2, 4, 8, 16, .......2^n $ શ્રેણીનો સમાંતર મધ્યક :
A
$\frac{{{2^n}\, - \,\,1}}{n}$
B
$\frac{{{2^{n\, + \,1}}\, - \,\,1}}{{n\,\, + \,\,1}}$
C
$\frac{{{2^n}\, - \,\,1}}{{n\,\, + \,\,1}}$
D
$\frac{{{2^{n\, + \,1}}\, - \,\,1}}{n}$
Solution
Given series is, $1,2,4,8,16, \ldots, 2^{ n }$
Clearly the series is in G.P with common ratio $2$ and number of terms is $n +1$
Thus sum of the series is $S =\frac{1-2^{n+1}}{1-2}=2^{ n +1}-1$
Hence A.M of this series is $=\frac{ S }{ n +1}=\frac{2^{ n +1}-1}{ n +1}$
Standard 11
Mathematics
