標題:
F.4 ..m2 (Binomial Theorem)
發問:
1.in the expansion of (3x^2 - 1)^n in ascending powers of x ,where n is a positive integer , the coefficient of x^4 is -189.find n.2.if the coefficient of x^2 in the expansion of (3 - x)^4 (1+ ax)^3 is 378,find a.3.given that (1 -2x )^m (1+x)^n =1 -8x +18x^2 + terms involving higher powersof x... 顯示更多 1.in the expansion of (3x^2 - 1)^n in ascending powers of x ,where n is a positive integer , the coefficient of x^4 is -189.find n. 2.if the coefficient of x^2 in the expansion of (3 - x)^4 (1+ ax)^3 is 378, find a. 3.given that (1 -2x )^m (1+x)^n =1 -8x +18x^2 + terms involving higher powers of x .find m and n. THX...!!
最佳解答:
1. The coefficient of x^4 term: (-1)^(2 + 1) + nC2 * (3)^2 = -189 (-1) * n!/(n - 2)!2! * 9 = -189 n!/(n - 2)!2! = 21 n!/(n - 2) = 42 n(n - 1) = 42 n^2 - n - 42 = 0 (n - 7)(n + 6) = 0 n = 7 or n = -6 (rejected) Hence, n = 7 2. (3 - x)^4 = 3^4 - 4C1*(3)^3*x + 4C2*-(3)^2*x^2 + …… = 81 - 108x + 54x^2 + …… (1 + ax)^3 = (1)^3 + 3*(1)^2*(ax) + 3*(1)*(ax)^2 + (ax)^3 = 1 + 3ax + 3a^2x^2 + …… The coefficient of x^2 in the expansion of (3 - x)^4 (1+ ax)^3: (81)(3a^2) + (-108)(3a) + 54 = 378 243a^2 - 324a - 324 = 0 3a^2 - 4a - 4 = 0 (3a + 2)(a - 2) = 0 a = -2/3 or a = 2 3. (1 - 2x)^m = (1)^m - mC1*(1)^(m - 1)*(2x) + mC2*(1)^(m - 2)*(2x)^2 + …… = 1 - 2mx + 2m(m - 1)x^2 + …… (1 + x)^n = (1)^n + nC1*(1)^(n - 1)*(x) + nC2-*(1)^(n - 2)*(x)^2 + …… = 1 + nx + [n(n - 1)/2]x^2 + …… x term of the expansion of (1 - 2x )^m (1 + x)^n: (1)(n) + (1)(-2m) = -8 n = 2m - 8 …… (1) x^2 term of the expansion of (1 - 2x )^m (1 + x)^n: (1)[n(n - 1)/2] + (-2m)(n) + (1)[2m(m - 1)] = 18 n(n - 1) - 4mn + 4m(m - 1) = 36 …… (2) Put (1) into (2): (2m - 8)(2m - 8 - 1) - 4m(2m - 8) + 4m(m - 1) = 36 4m^2 - 34 m + 72 - 8m^2 + 32m + 4m^2 - 4m = 36 -6m = -36 m = 6 Put m = 10 into (1): n = 2(6) - 8 n = 4 Hence, m = 6 and n = 4
其他解答:
F.4 ..m2 (Binomial Theorem)
發問:
1.in the expansion of (3x^2 - 1)^n in ascending powers of x ,where n is a positive integer , the coefficient of x^4 is -189.find n.2.if the coefficient of x^2 in the expansion of (3 - x)^4 (1+ ax)^3 is 378,find a.3.given that (1 -2x )^m (1+x)^n =1 -8x +18x^2 + terms involving higher powersof x... 顯示更多 1.in the expansion of (3x^2 - 1)^n in ascending powers of x ,where n is a positive integer , the coefficient of x^4 is -189.find n. 2.if the coefficient of x^2 in the expansion of (3 - x)^4 (1+ ax)^3 is 378, find a. 3.given that (1 -2x )^m (1+x)^n =1 -8x +18x^2 + terms involving higher powers of x .find m and n. THX...!!
最佳解答:
1. The coefficient of x^4 term: (-1)^(2 + 1) + nC2 * (3)^2 = -189 (-1) * n!/(n - 2)!2! * 9 = -189 n!/(n - 2)!2! = 21 n!/(n - 2) = 42 n(n - 1) = 42 n^2 - n - 42 = 0 (n - 7)(n + 6) = 0 n = 7 or n = -6 (rejected) Hence, n = 7 2. (3 - x)^4 = 3^4 - 4C1*(3)^3*x + 4C2*-(3)^2*x^2 + …… = 81 - 108x + 54x^2 + …… (1 + ax)^3 = (1)^3 + 3*(1)^2*(ax) + 3*(1)*(ax)^2 + (ax)^3 = 1 + 3ax + 3a^2x^2 + …… The coefficient of x^2 in the expansion of (3 - x)^4 (1+ ax)^3: (81)(3a^2) + (-108)(3a) + 54 = 378 243a^2 - 324a - 324 = 0 3a^2 - 4a - 4 = 0 (3a + 2)(a - 2) = 0 a = -2/3 or a = 2 3. (1 - 2x)^m = (1)^m - mC1*(1)^(m - 1)*(2x) + mC2*(1)^(m - 2)*(2x)^2 + …… = 1 - 2mx + 2m(m - 1)x^2 + …… (1 + x)^n = (1)^n + nC1*(1)^(n - 1)*(x) + nC2-*(1)^(n - 2)*(x)^2 + …… = 1 + nx + [n(n - 1)/2]x^2 + …… x term of the expansion of (1 - 2x )^m (1 + x)^n: (1)(n) + (1)(-2m) = -8 n = 2m - 8 …… (1) x^2 term of the expansion of (1 - 2x )^m (1 + x)^n: (1)[n(n - 1)/2] + (-2m)(n) + (1)[2m(m - 1)] = 18 n(n - 1) - 4mn + 4m(m - 1) = 36 …… (2) Put (1) into (2): (2m - 8)(2m - 8 - 1) - 4m(2m - 8) + 4m(m - 1) = 36 4m^2 - 34 m + 72 - 8m^2 + 32m + 4m^2 - 4m = 36 -6m = -36 m = 6 Put m = 10 into (1): n = 2(6) - 8 n = 4 Hence, m = 6 and n = 4
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