Prove that the co

To Prove : coefficient of xⁿ in (1+x)²ⁿ = 2 × coefficient of xⁿ in (1+x)^2n-1

For (1+x)²ⁿ,

a=1, b=x and m=2n

We have a formula,

To get the coefficient of xⁿ, we must have,

xⁿ = x^r

• r = n

Therefore, the coefficient of xⁿ

………

………..

………cancelling n

Therefore, the coefficient of xⁿ in (1+x)²ⁿ………eq(1)

Now for (1+x)^2n-1,

a=1, b=x and m=2n-1

We have formula,

To get the coefficient of xⁿ, we must have,

xⁿ = x^r

• r = n

Therefore, the coefficient of xⁿ in (1+x)^2n-1

…..multiplying and dividing by 2

Therefore,

coefficient of xⁿ in (1+x)^2n-1 = � × coefficient of xⁿ in (1+x)²ⁿ

or coefficient of xⁿ in (1+x)²ⁿ = 2 × coefficient of xⁿ in (1+x)^2n-1

Hence proved.