1.

If C_(0),C_(1),C_(2),...C_(n) are the binomial coefficients in the expansion of (1+x)^(n) then prove that:C_(0)+(C_(1))/(2)+(C_(2))/(3)+……+(C_(n))/(n+1)=(2^(n+1)-1)/(n+1)

Answer»


Solution :`L.H.S. =C_(0) +(C_(1))/(2)+(C_(2))/(3)+….+(C_(N))/(n+1)`
`=1+(n)/(2)+(n(n-1))/(|ul2.3)+....+(1)/(n+1)`
` =(1)/(n+1)[(n+1)+((n+2)n)/(|ul2)`
`+((n+1)n(n-1))/(|ul3)+.....+1]`
`=(1)/(n+1)[{1+(n+1)+((n+1)n)/(|ul2)`
`+((n+1)n(n-1))/(|ul3)+......+1}-]`
`=(1)/(n+1)[{.^(n+1)C_(0)+^(n+1)C_(1)+^(n+1)C_(2)`
`+^(n+1)C_(3)+......+^(n+1)C_(n+1)}-1]`
`=(1)/(n+1)[2^(n+1)-1]`
`=(2^(n+1)-1)/(n+1)=R.H.S.` Hence PROVED


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