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Find the sum of the coefficients of all the integral powers of `x` in the expansion of `(1+2sqrt(x))^(40)dot`

Text Solution

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`(1+2sqrt(x))^(40) = .^(40)C_(0)+.^(40)C_(1)2sqrt(x)+"....."+.^(40)C_(40)(2sqrt(x))^(40)`
and `(1-2sqrt(x))^(40)=.^(40)C_(0)-.^(40)C_(1)2sqrt(x)+"....."+.^(40)C_(40)(2sqrt(x))^(40)`
`:. (1+2sqrt(x))^(40) + (1-2sqrt(x))^(40)`
`= 2[.^(40)C_(0) + .^(40)C_(2)(2sqrt(x))^(2)+"....."+.^(40)C_(40)(2sqrt(x))^(40)]`
Putting `x = 1`, we get
`.^(40)C_(0) + .^(40)C_(2)(2)^(2) + "...." + .^(40)C_(40)(2)^(40) = (3^(40)+1)/(2)`
`:.` Sum of all the integral powers of `x = (3^(40)+1)/(2)`
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