Find the sum of the coefficients of two middle terms in the binomial expansion of

Question:

Find the sum of the coefficients of two middle terms in the binomial expansion of $(1+x)^{2 n-1}$.

Solution:

$(1+x)^{2 n-1}$

Here, $n$ is an odd number.

Therefore, the middle terms are $\left(\frac{2 n-1+1}{2}\right)^{\text {th }}$ and $\left(\frac{2 n-1+1}{2}+1\right)^{\text {th }}$, i. e.,$n^{\text {th }}$ and $(n+1)^{\text {th }}$ terms.

Now, we have

$T_{n}=T_{n-1+1}$

$={ }^{2 n-1} C_{n-1}(x)^{n-1}$

And,

$T_{n+1}=T_{n+1}$

$={ }^{2 n-1} C_{n}(x)^{n}$

$\therefore$ the coefficients of two middle terms are ${ }^{2 n-1} C_{n-1}$ and ${ }^{2 n-1} C_{n}$.

Now,

${ }^{2 \mathrm{n}-1} \mathrm{C}_{\mathrm{n}-1}+{ }^{2 \mathrm{n}-1} \mathrm{C}_{\mathrm{n}}={ }^{2 \mathrm{n}} \mathrm{C}_{\mathrm{n}}$

Hence, the sum of the coefficients of two middle terms in the binomial expansion of $(1+x)^{2 n-1}$ is ${ }^{2 n} C_{n}$.

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