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Asked by dineshchem108 | 6th Dec, 2018, 10:39: PM

Expert Answer:

begin mathsize 16px style S subscript n space equals space sum from r equals 1 to n of fraction numerator 1 plus 2 plus 2 squared........ plus 2 to the power of r minus 1 end exponent over denominator 2 to the power of r end fraction space space............... left parenthesis 1 right parenthesis
L e t space space space space X space equals space 1 plus 2 plus 2 squared................ plus 2 to the power of r minus 1 end exponent space equals space fraction numerator 2 to the power of r minus 1 over denominator 2 minus 1 end fraction space equals space 2 to the power of r space minus space 1 space space........... left parenthesis 2 right parenthesis
m a k i n g space s u b s t i t u t i o n space f r o m space e q n. left parenthesis 2 right parenthesis space i n space e q n. left parenthesis 1 right parenthesis
S subscript n space equals space sum from r equals 1 to n of open square brackets 1 space minus space 1 over 2 to the power of r close square brackets space equals space sum from r equals 1 to n of 1 space minus space sum from r equals 1 to n of 1 over 2 to the power of r space space equals space n space minus space 1 half open square brackets 1 plus 1 half plus 1 over 2 squared plus......1 over 2 to the power of n minus 1 end exponent close square brackets
S subscript n equals space n space minus space 1 half open square brackets fraction numerator 1 minus open parentheses begin display style 1 half end style close parentheses to the power of n over denominator 1 minus begin display style 1 half end style end fraction close square brackets space equals space n space minus 1 plus open parentheses 1 half close parentheses to the power of n space end style

Answered by Thiyagarajan K | 7th Dec, 2018, 02:49: AM

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