If the coefficients of (2r + 1)th term and (r + 2)th term in the expansion
Question:

If the coefficients of (2r + 1)th term and (r + 2)th term in the expansion of (1 + x)43 are equal, find r.

Solution:

Given : $-(1+x)^{43}$

We know that the coefficient of the $r$ th term in the expansion of $(1+x)^{n}$ is ${ }^{n} C_{r-1}$

Therefore, the coefficients of the $(2 r+1)$ th and $(r+2)$ th term $s$ in the given expression are ${ }^{43} C_{2 r+1-1}$ and ${ }^{43} C_{r+2-1}$

For these coefficients to be equal, we must have:

$\Rightarrow 2 r=r+1 \quad$ or, $2 r+r+1=43 \quad\left[\because{ }^{n} C_{r}={ }^{n} C_{s} \Rightarrow r=s\right.$ or $\left.r+s=n\right]$

$\Rightarrow r=14 \quad[\because$ for $r=1$ it gives the same term]

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