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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^(2))/(2) .^(10)C_(1) + (2^(3))/(3) .^(10)C_(2)+(2^(4))/(4).^(10)C_(3) + "...." + (2^(11))/(11).^(10)C_(10)`
` = underset(r=0)overset(10)sum(.^(10)C_(r))/(r+1)2^(r+1)=underset(r=0)overset(10)sum(.^(11)C_(r+1)2^(r+1))/(11)`
` = ((1+2)^(11)-1)/(11) = (3^(11)-1)/(11)`
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