Show that the coefficient of the middle term in the expansion of `(1+x)^(2n)`is equal to the sum of the coefficients of two middle terms in the expansion of `(1+x)^(2n-1)`.

Explanation

To solve the problem, we need to find the coefficient of the middle term in the expansion of \((1+x)^{2n}\) and show that it is equal to the sum of the coefficients of the two middle terms in the expansion of \((1+x)^{2n-1}\).

### Step 1: Find the middle term in the expansion of \((1+x)^{2n}\)

The total number of terms in the expansion of \((1+x)^{2n}\) is \(2n + 1\). Therefore, the middle term is the \((n+1)\)-th term.

Using the binomial theorem, the general term \(T_k\) in the expansion of \((1+x)^{n}\) is given by:
\[
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