Rút gọn tổng: \(S=C\overset{1}{n}+1.2C\overset{2}{n}+2.3C\overset{3}{n}+...+\left(n-1\right)nC\overset{n}{n}\) bằng:
A. \(\left(n-1\right)n.2^{n-2}\)
B. \(n.2^{n-2}\)
C. \(\left(n-1\right)n.2^{n-1}+n\)
D. \(\left(n-1\right)n.2^{n-2}+n\)
Rút gọn tổng: \(S=C\overset{1}{n}+1.2C\overset{2}{n}+2.3C\overset{3}{n}+...+\left(n-1\right)nC\overset{n}{n}\) bằng:
A. \(\left(n-1\right)n.2^{n-2}\)
B. \(n.2^{n-2}\)
C. \(\left(n-1\right)n.2^{n-1}+n\)
D. \(\left(n-1\right)n.2^{n-2}+n\)
Rút gọn tổng: \(S=C\overset{0}{n}+C\overset{1}{n}+2.C\overset{2}{n}+...+nC\overset{n}{n}\) bằng:
A. \(n.2^n+1\)
B. \(2^n+1\)
C. \(n.2^{n-1}+1\)
D. \(n.2^{n+1}\)
Xét khai triển:
\(\left(1+x\right)^n=C_n^0+xC_n^1+x^2C_n^2+...+x^nC_n^n\)
Đạo hàm 2 vế:
\(n\left(1+x\right)^{n-1}=C_n^1+2xC_n^2+...+n.x^{n-1}C_n^n\)
Thay \(x=1\)
\(\Rightarrow n.2^{n-1}=C_n^1+2C_n^2+...+nC_n^n\)
\(\Rightarrow n.2^{n-1}+1=C_n^0+C_n^1+2C_n^2+...+nC_n^n\)
\(\Rightarrow S=n.2^{n-1}+1\)
Rút gọn tổng: \(S=-C\overset{1}{2019}+1.2C\overset{2}{2019}-2.3C\overset{3}{2019}+...+2017.2018C\overset{2018}{2019}-2018.2019C\overset{2019}{2019}\) bằng:
A. 1
B.. 2019
C. 0
D. -2019
Rút gọn biểu thức \(S\left(x\right)=\dfrac{1}{x^2}+\dfrac{2}{x^3}+\dfrac{3}{x^4}+...+\dfrac{n}{x^{n+1}}\) bằng:
A. \(S=\dfrac{x^{n+1}-\left(n+1\right)x+n}{x^{n+1}\left(x-1\right)^2}\)
B. \(S=\dfrac{x^{n+1}-\left(n+1\right)x+n}{x^{2n}\left(x-1\right)^2}\)
C. \(S=\dfrac{x^n-\left(n+1\right)x+n}{x^n\left(x-1\right)^2}\)
D. \(S=\dfrac{x^{n+1}-\left(n+1\right)x+n}{x^n\left(x-1\right)^2}\)
Rút gọn biểu thức \(S\left(x\right)=\dfrac{1}{x^2}+\dfrac{2}{x^3}+\dfrac{3}{x^4}+...+\dfrac{n}{x^{n+1}}\) bằng:
A. \(S=\dfrac{x^{n+1}-\left(n+1\right)x+n}{x^{n+1}\left(x-1\right)^2}\)
B. \(S=\dfrac{x^{n+1}-\left(n+1\right)x+n}{x^{2n}\left(x-1\right)^2}\)
C. \(S=\dfrac{x^n-\left(n+1\right)x+n}{x^n\left(x-1\right)^2}\)
D. \(S=\dfrac{x^{n+1}-\left(n+1\right)x+n}{x^n\left(x-1\right)^2}\)
\(S\left(x\right)=\dfrac{1}{x^2}+\dfrac{2}{x^3}+...+\dfrac{n}{x^{n+1}}\)
\(\Rightarrow x.S\left(x\right)=\dfrac{1}{x}+\dfrac{2}{x^2}+\dfrac{3}{x^3}+...+\dfrac{n}{x^n}\)
\(\Rightarrow x.S\left(x\right)-S\left(x\right)=\dfrac{1}{x}+\dfrac{1}{x^2}+\dfrac{1}{x^3}+...+\dfrac{1}{x^n}-\dfrac{n}{x^{n+1}}\)
\(\Rightarrow\left(x-1\right)S\left(x\right)=\dfrac{1}{x}.\dfrac{1-\left(\dfrac{1}{x}\right)^n}{1-\dfrac{1}{x}}-\dfrac{n}{x^{n+1}}=\dfrac{x^n-1}{x^n\left(x-1\right)}-\dfrac{n}{x^{n+1}}=\dfrac{x^{n+1}-x-n\left(x-1\right)}{x^{n+1}\left(x-1\right)}\)
\(\Rightarrow S\left(x\right)=\dfrac{x^{n+1}-\left(n+1\right)x+n}{x^{n+1}\left(x-1\right)^2}\)
Rút gọn tổng \(S=C\overset{1}{2019}-2C\overset{2}{2019}+...-2018C\overset{2018}{2019}+2019C\overset{2019}{2019}\) bằng:
A. 2019
B.1
C. -2019
D. 0
Viết phương trình hóa học của các phản ứng biểu diễn các chuyển đổi sau:
S \(\overset{\left(1\right)}{->}\)H\(_2\)S \(\overset{\left(2\right)}{->}\) SO\(_2\)\(\overset{\left(3\right)}{->}\)SO\(_3\)\(\overset{\left(4\right)}{->}\)H\(_2\)SO\(_4\)
Trong phản ứng trên phản ứng nào là phản ứng oxi hóa- khử
Help me!!!
S + H2 -to-> H2S
2H2S + 3O2 -to-> 2SO2 + 2H2O SO2 + 1/2O2 -to-> SO3 SO3 + H2O => H2SO4 Các phản ứng oxi hóa khử : (1) , (2) , (3)rút gọn biểu thức
\(D=\left(n+1\right)\left(n^2+1\right)\left(n^{2^2}+1\right)\left(n^{2^3}+1\right)...\left(n^{2^m}+1\right)\)
rút gọn các phân thức vs n là số tự nhiên:
a,\(\frac{\left(n+1\right)!}{n!\left(n+2\right)}\)b,\(\frac{n!}{\left(n+1\right)!-n!}\)c,\(\frac{\left(n+1\right)!-\left(n+2\right)!}{\left(n+1\right)+\left(n+2\right)!}\)
a) \(\frac{\left(n+1\right)!}{n!\left(n+2\right)}=\frac{n!\left(n+1\right)}{n!\left(n+2\right)}=\frac{n+1}{n+2}\)
b)\(\frac{n!}{\left(n+1\right)!-n!}=\frac{n!}{n!\left(n+1\right)-n!}=\frac{n!}{n!\left(n+1-1\right)}=\frac{1}{n}\)
c)\(\frac{\left(n+1\right)!-\left(n+2\right)!}{\left(n+1\right)!+\left(n+2\right)!}=\frac{n!\left(n+1\right)-n!\left(n+1\right)\left(n+2\right)}{n!\left(n+1\right)+n!\left(n+1\right)\left(n+2\right)}=\frac{n!\left(n+1\right)\left(1-n-2\right)}{n!\left(n+1\right)\left(1+n+2\right)}=\frac{-n-1}{n+3}\)
( Kí hiệu n!=1.2.3.4...n)