Consider the expansion of $(1 + x)^{2 n + 1}$
If the coefficients of $x^{r}$ and $x^{r + 1}$ are equal in the expansion, then $r$ is equal to

n

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\frac{2 n - 1}{2}

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\frac{2 n + 1}{2}

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n + 1

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Solution

The correct option is A $n$
We have, $(1 + x)^{2 n + 1}$
General term, $T_{r + 1} = C_{r}^{2 n + 1} x^{r}$
we have, $C_{r}^{2 n + 1} = C_{r + 1}^{2 n + 1}$
Solving this, we get
$2 n + 1 = r + r + 1$
$\Rightarrow 2 r = 2 n$
$\Rightarrow r = n$
Hence, A is the correct option.

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