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標題:

F.4 (x+1)^29

發問:

Given that (x+1)^29, the coefficient of R th term and the coefficient of (R+1)th term,the ratio is 3:2,find the value of R.

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(x + 1)^29 = x^29 + C(29, 1)x^28 + C(29, 2)x^27 + ...... + C(29, 28)x +1 Coefficient of the Rth term = C(29, R-1) Coefficient of the (R+1)th term = C(29, R) C(29, R-1) : C(29, R) = 3 : 2 29!/(R-1)![29-(R-1)]! : 29!/R!(29-R)! = 3 : 2 1/(R-1)!(30-R)! : 1/R!(29-R)! = 3 : 2 1/(R-1)!(30-R)(29-R)! : 1/R(R-1)!(29-R)! = 3 : 2 1/(30-R) : 1/R = 3 : 2 2 * [1/(30-R)] = 3 * (1/R) 2/(30-R) = 3/R 2R = 3(30-R) 2R = 90-3R 5R = 90 R = 18

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