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

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Solution

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

Here, n is an odd number.

Therefore, the middle terms are $(\frac{2 n - 1 + 1}{2}) t h$ and $(\frac{2 n - 1 + 1}{2} + 1) t h$ , i.e., n th and (n+1)th terms.

Now, we have

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

And,

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

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

Now,

$^{2 n - 1} C_{n - 1} +^{2 n - 1} C_{n} =^{2 n} C_{n}$

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

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Find the sum of the coefficients of two middle terms in the binomial expansion of (1+x)2n−1

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