1.

Let S = sum _(r=1)^(30) (""^(30+r)C_(r) (2r-1))/(""^(30)C_(r)(30+r)),K=sum_(r=0)^(30) (""^(30)C_(r))^(2) and G=sum_(r=0)^(60) (-1)^(r)(""^(60)C_(r) )^(2) The value fo (G-S)is

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`2^(30)`
`2^(60)`

Solution :`because S = sum _(R=1)^(30) (""^(30+r)C_(r) (2r-1))/(""^(30)C_(r)(30+r))=sum_(r=0)^(30) (""^(30+r)C_(r))/(""^(30)C_(r))(1-(30-r+1)/(30+r))`
`=sum_(r=0)^(30)[ (""^(30+r)C_(r))/(""^(30)C_(r)) -(""^(30+r)C_(r))/(""^(30)C_(r))cdot ((30-r+1))/((30+r))]`
`=sum_(r=0)^(30)[ (""^(30+r)C_(r))/(""^(30)C_(r)) -""^((30+r)/(r)cdot ^(29+r)C_(r-1))/(""^(30)C_(r))cdot ((30-r+1))/(30+r)]`
`=sum_(r=0)^(30)[ (""^(30+r)C_(r))/(""^(30)C_(r)) -( ""^(29+r)C_(r-1))/(""^(30)C_(r-1)) ][because (""^(n)C_(r))/(""^(n)C_(r-1))=(n-r+1)/r]`
For n = 30 `((31-r)/rcdot ""^(30)C_(r)=""^(30)C_(r-1))`
`=(""^(30+30)C_(30))/ (""^(30)C_(30)) - (""^(29-1)C_(0))/(""^(30)C_(0))= ""^(60)C_(30)-1`
`K = sum _(r=1) ^(30) (""^(30)C_(r))^(2) = ""^(60)C_(30) and G = sum_(r=0)^(60) (-1)^(r) (""^(60)C_(r))^(2)`
`(""^(60)C_(0))^(2) - (""^(60)C_(1))^(2)+(""^(60)C_(2))^(2)-...+(""^(60)C_(60))=""^(60)C_(30)`
[`because n=60` is even ]
`G-S = "" ^(60)C_(1) - ("" ^(60)C_(30)-1) =1`


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