Question

Find the sum of the coefficient of to middle terms in the binomial expansion of `(1+x)^(2n-1)`

Answer 1

Here, the Binominal expansion is `(1 + x)^(2n - 1)`
Since, this expansion has two middle term i.e., `((2n - 1 + 1)/(2))th " term and " ((2n - 1 + 1)/(2) + 1) th` term i.e., `nth` term and `(n + 1)` th term.
`:.` Coefiicient of `nth` term `= .^(2n - 1)C_(n -1)`
Coefficient of `(n + 1)` th term `= .^(2n - 1)C_(n)`
Sum of coefficients `= .^(2n - 1)C_(n -1) + .^(2n -1)C_(n)`
`= .^(2n - 1 + 1)C_(n) = .^(2n)C_(n) " " [:. .^(n)C_(r) + .^(n)C_(r - 1) = .^(n + 1)C_(r)]`