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Given positive integers r>1,n> 2, n bei...

Given positive integers `r>1,n> 2, n` being even and the coefficient of `(3r)th` term and `(r+ 2)th` term in the expansion of `(1 +x)^(2n)` are equal; find r

A

`r=n/2`

B

`n = 3r`

C

`n = 2r + 1`

D

None of these

Text Solution

Verified by Experts

The correct Answer is:
A
Given that, `r gt 1, n gt 2` and the coefficients of `(3r) th " and " (r + 2)th` term are equal in the expansion of `(1 + x)^(2n)`
Then, `T_(3r) = T_(3r - 1 + 1) = .^(2n)C_(3r - 1) x^(3r - 1)`
and `T_(r + 2) = T_(r + 1 + 1) = .^(2n)C_(r + 1) x^(r + 1)`
Given, `.^(2n)C_(3r - 1) = .^(2n)C_(r + 1) " " [:. .^(n)C_(x) = .^(n)C_(y) rArr x + y = n]`
`rArr 3 r - 1 + r + 1 = 2n`
`rArr 4 r = 2n rArr n = (4r)/(2)`
`.: n = 2r`
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