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Find the sum 2..^(10)C(0) + (2^(2))/(...

Find the sum
`2..^(10)C_(0) + (2^(2))/(2).^(10)C_(1) + (2^(3))/(3).^(10)C_(2)+(2^(4))/(4).^(10)C_(3)+"...."+(2^(11))/(11).^(10)C_(10)`.

Text Solution

Verified by Experts

`2^(10)C_(0)+(2.^(210)C_(1))/(2)+(2^(3).^(10)C_(2))/(3)+ . . . .+(2^(11).^(10)C_(10))/(11)`
`=2^(10)C_(0)+(2^(2))/(2).^(10)C_(1)+(2^(3))/(3).^(10)C_(2)+ . . . .+(2^(1))/(11).^(10)C_(10)`
`=[(2lfloor(10))/(lfloor(0)floor(10))+(2^(2))/(2)(lfloor(10))/(lfloor(9)lfloor(1))+(2^(3))/(3)(lfloor(10))/(lfloor(8)lfloor(2))+ . . .+(2^(11)lfloor(10))/(11lfloor(0)lfloor(10))]`
`=(1)/(11)[(2.11.lfloor(10))/(lfloor(10)lfloor(0))+(2^(2).11.lfloor(10))/(lfloor(9)[2(1)])+ . . . .+(2^(11).11.lfloor(10))/(lfloor(0)(11lfloor(10)))]`
`=(1)/(11)[2.^(11)C_(1)+2^(2).^(11)C_(2)+2^(3).^(11)C_(3)+ . . . +2^(11).^(11)C_(11)]`
`=(1)/(11)[(1+2)^(11)-^(11)C_(0)]=(3^(11)-1)/(11)`
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