prove that the coefficient of `x^n` in the expansion of `(1+x)^(2n)` is twice the coefficient of `x^n` in the expansion of `(1+x)^(2n-1)`

Answer

N/a

Explanation

`(1+x)^(2n)`
`T_(r+1)+^(2n)C_(r)x^(r)`
`:.` Coefficient of `x^(n) " is " P=^(2n)C_(n).`
In `(1+x)^(2n-1)`
`T_(r+1)=^(2n-1)C_(r).x^(r) `
`:.` coefficient of `x^(n) " is " Q =^(2n-1) C_(n)`
To prove P=2Q
`" Now " (P)/(Q) =(.^(2n)C_(n))/(.^(2n-1)C_(n))=((2n)!n!(n-1)!)/(n!n!(2n-1)!)=(2n)/(n)=2`
`rArr " "P=2Q` Hence Proved.