If the coefficient of the middle term in the expansion of 1- x 2 n -2 is - and the coefficients of middle terms in the expansion of 1- x 2 n -1 are - and Y- then relation between - - and Y isA- -0B- - YC- -D- -

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Middle Terms

If the coeffi...

Question

If the coefficient of the middle term in the expansion of $(1 + x)^{2 n + 2}$ is $α$ and the coefficients of middle terms in the expansion of $(1 + x)^{2 n + 1}$ are $β$ and $γ$ , then relation between $α, β$ and $γ$ is-

α + β + γ = 0

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α = β + γ

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β = α + γ

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γ = β + α

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Solution

The correct option is B $α = β + γ$
$(n + 2)^{t h}$ term is the middle term in the expansion of $(1 + x)^{2 n + 2}$ .
Therefore, $α =^{2 n + 2} C_{n + 1} .$
$(n + 1)^{t h}$ and $(n + 2)^{t h}$ terms are middle terms in the expansion of $(1 + x)^{2 n + 1}$
Therefore, $β =^{2 n + 1} C_{n}$ and $γ =^{2 n + 1} C_{n + 1}$
Also,
$^{2 n + 1} C_{n} +^{2 n + 1} C_{n + 1} =^{2 n + 2} C_{n + 1}$
$(∵^{n} C_{r} +^{n} C_{r + 1} =^{n + 1} C_{r + 1})$
$\Rightarrow β + γ = α$

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Middle Terms

Standard XII Mathematics

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If the coefficient of the middle term in the expansion of (1+x)2n+2 is α and the coefficients of middle terms in the expansion of (1+x)2n+1 are β and γ, then relation between α,β and γ is-

If the coefficient of the middle term in the expansion of $(1 + x)^{2 n + 2}$ is $α$ and the coefficients of middle terms in the expansion of $(1 + x)^{2 n + 1}$ are $β$ and $γ$ , then relation between $α, β$ and $γ$ is-