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If some three consecutive coefficeints i...

If some three consecutive coefficeints in the binomial expanison of `(x + 1)^(n)` in powers of x are in the ratio 2 : 15 : 70, then the average of these three coefficients is

A

232

B

625

C

964

D

227

Text Solution

Verified by Experts

The correct Answer is:
A
`.^(n)C_(r - 1) , .^(n)C_(r + 1) = 2 : 15 : 70`, `.^(n)C_(r - 1): .^(n)C_(r ) = (2)/(15), (.^(n)C_(r ))/(.^(n)C_(r - 1)) = (15)/(2)`
`(n - r + 1)/(r ) = (15)/(2)`, `2n - 2r + 2 + 15 r`
`2n - 17r = - 2`
`(.^(n)C_(r + 1))/(.^(n)C_(r )) = (70)/(15) implies (n - (r + 1) + 1)/(r + 1) = (14)/(3)`
`(n - r)/(r + 1) = (14)/(3) implies 3n - 3r = 14 r + 14`
`3n - 17 r = 14`
`2n - 17 r = - 2`
`overline(n = 16)`
`48 - 17r = 14`
`17r = 34` `implies` `r = 2`
Average `= (.^(16)C_(1) + .^(16)C_(2) + .^(16)C_(3))/(3) = (16 + .^(17)C_(3))/(3) = (16 + (17 .16.15)/(1.2.3))/(3) = (16 + 680)/(3) = (696)/(3) = 232`
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