Question

Find the sum of the last 30 coefficients in the expansion of `(1+x)^(59),` when expanded in ascending powers of `xdot`

Answer 1

Correct Answer - `2^(58)`
There are 60 terms is the expansion of `(1+x)^(59)`. Sum of last `30` coefficient is
`S = .^(59)C_(30) + .^(59)C_(31) + "….." + .^(59)C_(58) + .^(59)C_(59)`
`:. S = .^(59)C_(29) + .^(59)C_(28) + "……." + .^(59)C_(1) + .^(59)C_(0)` [Using `.^(n)C_(r ) = .^(n)C_(n-r)`]
Adding the above two expansions, we get
`2S = .^(59)C_(0) + .^(59)C_(1) + "......" + .^(59)C_(59) = 2^(59)`
or `S = 2^(58)`