Question

a, tính tổng sau S=\(C^1_{14}-2C^2_{14}+3C^2_{14}-......-14C^{14}_{14}\)         ^{_{b, S=\(9.2^8C^0_9-8.2^7C^1_9+7.2^6C^2_9-.......+C^8_9\)}}

Answer 1

a.

Xét khai triển:

\(\left(1+x\right)^{14}=C_{14}^0+C_{14}^1x+...+C_{14}^{14}x^{14}\)

Đạo hàm 2 vế:

\(14\left(1+x\right)^{13}=C_{14}^1+2C_{14}^2x+...+14C_{14}^{14}x^{13}\)

Cho \(x=-1\) ta được:

\(0=C_{14}^1-2C_{14}^2+...-14C_{14}^{14}\)

\(\Rightarrow S=0\)

b. Xét khai triển:

\(\left(1+2x\right)^9=C_9^0+C_9^1\left(2x\right)+C_9^2\left(2x\right)^2+...+C_9^9\left(2x\right)^9\)

\(=C_9^9+C_9^8\left(2x\right)+C_9^7\left(2x\right)^2+...+C_9^0\left(2x\right)^9\)

Đạo hàm 2 vế:

\(18\left(1+2x\right)^8=2C_9^8+2.2^3C_9^7x+3.2^4C_9^6x^2+...+9.2^9C_9^0x^8\)

\(\Rightarrow9\left(1+2x\right)^8=C_9^8+2.2^2C_9^7x+...+9.2^8C_9^0x^8\)

Cho \(x=-1\)

\(\Rightarrow9=C_9^8-2.2^2C_9^7+...+9.2^8C_9^0\)

\(\Rightarrow S=9\)