Q. 283.7( 4 Votes )

# 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 :

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}

Hence L.H.S = R.H.S.

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