The general term T_{r+1} in the binomial expansion is given by T_{r+1} = ^{n}C_{r} a^{n-r} b^{r}

The general term for binomial (1+x)^{2n} is

T_{r+1} = ^{2n}C_{r} x^{r} …………………..1

To find the coefficient of x^{n}

r = n

T_{n+1} = ^{2n}C_{n} x^{n}

The coefficient of x^{n} = ^{2n}C_{n}

The general term for binomial (1+x)^{2n-1} is

T_{r+1} = ^{2n-1}C_{r} x^{r}

To find the coefficient of x^{n}

Putting n = r

T_{r+1} = ^{2n-1}C_{r} x^{n}

The coefficient of x^{n} = ^{2n-1}C_{n}

We have to prove

Coefficient of x^{n} in (1+x)^{2n} = 2 coefficient of x^{n} in (1+x)^{2n-1}

Consider LHS = ^{2n}C_{n}

= \( \dfrac{2!}{n!(2n-n)!}\)

= \( \dfrac{2n!}{n!n!}\)

Consider RHS = \(2 \times ^{2n-1}C_n\)

= \(2\times \dfrac{(2n-1)!}{n!(2n-1-n)!}\)

= \( 2\times \dfrac{(2n-1)!}{n!(n-1)!}\)

Now multiplying and dividing by n we get

= \( 2\times \dfrac{(2n-1)!}{n!(n-1)!}\times \dfrac{n}{n}\)

= \( \dfrac{2n(2n-1)!}{n!n(n-1)!}\)

From above equations LHS = RHS

Hence the proof

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