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_(0)+(C_(1))/(2)+(C_(2))/(3)+...+(C_(n))/(n+1)=(2^(n+1)-1)/(n+1)" or "sum+(k=0)^(n)(C_(k))/(k+1)=(2^(n+1)-1)/(n+1)` |
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Answer» दिया गया श्रेणी है : `C_(0)+(C_(1))/(2)+(C_(2))/(3)+...+(C_(n))/(n+1)` r वाँ पद `t_(r)=(""^(n)C_(r-1))/(r)` अब, `C_(0)+(C_(1))/(2)+(C_(2))/(3)+...+(C_(n))/(n+1)` `=underset(r=1)overset(n+1)sum(""^(n)C_(r-1))/(r)=underset(r=1)overset(n+1)sum(""^(n+1)C_(r))/(n+1)[because(""^(n)C_(r-1))/(r)=(""^(n+1)C_(r))/(n+1)]` `=(1)/(n+1)(""^(n+1)C_(1)+""^(n+1)C_(2)+...+""^(n+1)C_(n+1))` `=(1)/(n+1)(""^(n+1)C_(0)+""^(n+1)C_(1)+""^(n+1)C_(2)+...+""^(n+1)C_(n+1)-""^(n+1)C_(0))` `=(1)/(n+1)(2^(n+1)-1)=(2^(n+1)-1)/(n+1)" "[because""^(n+1)C_(0)=1]` |
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