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F.4 A. Maths

發問:

1. Show that (1+ax+bx^2)^n = 1+nax+(nb+nC2a^)x^2+ (2 nC2ab+ nC3a^3)x^3 + terms involving higher powers of x, where n is a positive integer greater than 3. (b) If (1+ax+bx^2)^n = 1+7x-7x^2 + terms involving higher powers of x, where a and b are integers, find the values od a, b and n.

最佳解答:

1 (1+ax+bx^2)^n =[1+x(a+bx)]^n =1+n[x(a+bx)]+nC2 [x(a+bx)]^2+ nC3 [x(a+bx)]^3 +... =1+nax+nbx^2+nC2[x^2(a^2+2abx+...]+nC3[x^3(a^3+...] =1+nax+nbx^2+[nC2 a^2x^+2nC2 abx^3+...]+[nC3x^3+...] =1+nax+(nb+nc2a^2)x^2+(2nC2 ab+nC3a^3)x^3+... (b) na=7 (nb+nC2a^2)=-7 consider na=7 n is a positive integer greater than 3,a is an integer so n=7,a=1 put n=7, a=1 into(nb+nC2a^2)=-7 7b+7C2 a^2=-7 7b+21=-7 b=-4

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