Question

Find the sum of the last 30 coefficients in the expansion of `(1+x)^(59),` when expanded in ascending powers of `xdot`
A. `2^(58)`
B. `2^(29)`
C. `2^(28)`
D. `2^(59) - 2^(29)`

Answer 1

Correct Answer - a
We have
`(1 + x)^(59) = sum_(r = 0)^(29) ""^(59)C_(r) x^(r)`
`rArr 2^(59)= sum_(r = 0)^(29) ""^(59)C_(r)` [Putting x = 1]
`rArr 2^(59)= 2(sum_(r = 0)^(29) ""^(59)C_(r)) (because ""^(59)C_(r)= ""^(59)C_(59-r), r = 0,1,...29]`
`rArr 2^(58)= sum_(r = 0)^(29) ""^(59)C_(r)`
`rArr 2^(58)= sum_(r = 0)^(29) ""^(59)C_(59-r = 2^(58))` .