Xét khai triển: 

\(\left(x^2-1\right)^{20}=C_{20}^0-C_{20}^1.x^2+C_{20}^2x^4-...+C_{20}^{20}x^{20}\)

Thay \(x=2\)

\(\Rightarrow3^{20}=C_{20}^0-2^2C_{20}^1+2^4C_{20}^2-...+2^{40}C_{20}^{20}\)

\(\Rightarrow J=3^{20}\)

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 \(x=-1\)

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

\(\Rightarrow S=9\)

 \(S=C_{100}^1-C_{100}^2+...-C_{100}^{100}\)

Ta có:

\(\Rightarrow S_1=C_{100}^0-C_{100}^1+C_{100}^2+...+C_{100}^{100}=0\)

\(\Rightarrow C_{100}^0=C_{100}^1-C_{100}^2+...-C_{100}^{100}=1\)(chuyển vế)

Vậy \(S=1\)

a)Quy đồng: \(\frac{5}{8}=\frac{5.3}{8.3}=\frac{15}{24}\)

Vì \(\frac{5}{24}< \frac{10+5}{24}=\frac{15}{24}\)

\(\Rightarrow\frac{5}{24}< \frac{5+10}{24}=\frac{5}{8}\)

b) Quy đồng:

\(\frac{4}{9}=\frac{4.6}{9.6}=\frac{24}{9.6}\)

\(\frac{2}{3}=\frac{2.18}{3.18}=\frac{36}{9.6}\)

Vì \(\frac{36}{9.6}>\frac{24}{9.6}>\frac{6+9}{9.6}\)

\(\Rightarrow\frac{2}{3}>\frac{4}{9}>\frac{6+9}{6.9}\)

Xét khai triển:

\(\left(x+1\right)^n=C_n^0+C_n^1x+C_n^2x^2+...+C_n^nx^n\)

\(\Leftrightarrow x\left(x+1\right)^n=C_n^0.x+C_n^1x^2+C_n^2x^3+...+C_n^nx^{n+1}\)

Thay \(n=2000\) ta được:

\(x\left(x+1\right)^{2000}=C_{2000}^0x+C_{2000}^1x^2+C_{2000}^2x^3+...+C_{2000}^{2000}x^{2001}\)

Đạo hàm 2 vế:

\(\left(x+1\right)^{2000}+2000x\left(x+1\right)^{1999}=C_{2000}^0+2C_{2000}^1x+...+2001C_{2000}^{2000}x^{2000}\)

Thay \(x=1\) ta được:

\(2^{2000}+2000.2^{1999}=C_{2000}^0+2C_{2000}^1+...+2001.C_{2000}^{2000}\)

\(\Rightarrow S=2^{1999}\left(2+2000\right)=2002.2^{1999}\)

1=(2n+1)C0, (2n+1)Cn=(2n+1)C(n+1)...

2015