InterviewSolution
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InterviewSolution
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यदि `(1+x)^(n)=C_(0)+C_(1)x+C_(2)x^(2)+...+C_(n)x^(n)` तो साबित कीजिए कि `(C_(1))/(2)+(C_(3))/(4)+(C_(5))/(6)+(C_(7))/(8)+......=(2^(n))/(n+1)` |
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Answer» दिया गया श्रेणी है : `(C_(1))/(2)+(C_(3))/(4)+(C_(5))/(6)+......` इसका r वाँ पद `t_(r)=(""^(n)C_(2r-1))/(2r)=(""^(n)C_(2r-1))/((2r-1)+1)=(""^(n+1)C_(2r))/(n+1)` अब `(C_(1))/(2)+(C_(3))/(4)+(C_(5))/(6)+...` `=underset(r=1)sumt_(r)=(1)/(n+1)underset(r=1)sum""^(n+1)C_(2r)` `=(1)/(n+1)[""^(n+1)C_(2)+""^(n+1)C_(4)+""^(n+1)C_(6)+...]` `=(1)/(n+1)[(""^(n+1)C_(0)+""^(n+1)C_(2)+""^(n+1)C_(4)+.......)-""^(n+1)C_(0)]` `=(1)/(n+1)(2^(n)-1)" "[because""^(n)C_(0)+""^(n)C_(2)+""^(n)C_(4)+...=2^(n-1)]` `=(2^(n)-1)/(n+1)` |
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