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If the coefficient of the middle term in the expansion of $(1+x)^{2 n+2}$ is p and the coefficients of middle terms in the expansion of $(1+x)^{2 n+1}$ are q and r, then

Option: 1
$p+q=r$

Option: 2
$p+r=q$

Option: 3
$p=q+r$

Option: 4
$p+q+r=0$

Answers (1)

best_answer

Since $(n+2)^{\text {th }}$ term is the middle term in the expansion of $(1+x)^{2 n+2}$ , therefore $p={ }^{2 n+2} C_{n+1}$ . Since $(n+1)^{\text {th }}$ and $(n+2)^{\text {th }}$ terms are middle terms in the expansion of $(1+x)^{2 n+1}$ , therefore $q={ }^{2 n+1} C_n$ and $r={ }^{2 n+1} C_{n+1}$

But ${ }^{2 n+1} C_n+{ }^{2 n+1} C_{n+1}={ }^{2 n+2} C_{n+1}$

$\therefore q+r=p$

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vinayak

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