\(A=\left(1-\frac{1}{2}\right)\left(1-\frac{1}{3}\right)\left(1-\frac{1}{4}\right)...\left(1-\frac{1}{2003}\right)\left(1-\frac{1}{2004}\right)\)

\(A=\frac{1}{2}.\frac{2}{3}.\frac{3}{4}...\frac{2002}{2003}.\frac{2003}{2004}=\frac{1}{2004}\)

1/2004

\(A=\left(1-\frac{1}{2}\right)\cdot\left(1-\frac{1}{3}\right)...\left(1-\frac{1}{2004}\right)\)

\(\Rightarrow A=\frac{1}{2}\cdot\frac{2}{3}\cdot\frac{3}{4}...\frac{2002}{2003}\cdot\frac{2003}{2004}\)

\(\Rightarrow A=\frac{1\cdot2\cdot3...2003}{2\cdot3\cdot4...2004}=\frac{1}{2004}\)

B=1/2+1/2^2+1/2^3+...+1/2^2004

2B=1+1/2+1/2^2+...+1/2^2003

2B-B=(1+1/2+1/2^2+...+1/2^2003)-(1/2+1/2^2+1/2^3+...+1/2004)

2B-B=1+1/2+1/2^2+...+1/2^2003-1/2-1/2^2-1/2^3-...-1/2^2004 (TỐI GIẢN CÁC PHÂN SỐ GIỐNG NHAU)

B=1-1/2^2004

\(B=\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+...+\frac{1}{3^{2004}}+\frac{1}{3^{2005}}\)

\(\Rightarrow3B=1+\frac{1}{3}+\frac{1}{3^2}+...+\frac{1}{3^{2003}}+\frac{1}{3^{2004}}\)

\(\Rightarrow3B-B=1-\frac{1}{3^{2005}}\)

\(\Rightarrow2B=1-\frac{1}{3^{2005}}\)

\(\Rightarrow B=\frac{1-\frac{1}{3^{2005}}}{2}\)

Giải:

\(B=\dfrac{1}{3}+\dfrac{1}{3^2}+\dfrac{1}{3^3}+...+\dfrac{1}{3^{2004}}+\dfrac{1}{3^{2005}}\)

\(\Leftrightarrow\dfrac{1}{3}B=\dfrac{1}{3^2}+\dfrac{1}{3^3}+\dfrac{1}{3^4}...+\dfrac{1}{3^{2005}}+\dfrac{1}{3^{2006}}\)

\(\Leftrightarrow B-\dfrac{1}{3}B=\dfrac{1}{3}-\dfrac{1}{3^{2006}}\)

\(\Leftrightarrow\dfrac{2}{3}B=\dfrac{1}{3}-\dfrac{1}{3^{2006}}\)

\(\Leftrightarrow B=\dfrac{\dfrac{1}{3}-\dfrac{1}{3^{2006}}}{\dfrac{2}{3}}\)

\(\Leftrightarrow B=\dfrac{1-\dfrac{1}{3^{2005}}}{2}\)

\(\Leftrightarrow B=\dfrac{\dfrac{3^{2005}-1}{3^{2005}}}{2}\)

\(\Leftrightarrow B=\dfrac{3^{2005}-1}{2.3^{2005}}\)

Vậy ...

Chúc bn học tốt!

Đặt B = 2004+2003/2+2002/3+...+1/2004 B có 2004 phân số tách số 2004 = 1+1+1+...+1(2004 số 1) ghép 2004 số 1 vào từng nhóm như sau: B=(1+ 2003/2)+ (1+ 2002/3)+...+(1+1/2004) +1 B = 2005/2+2005/3+......+2005/2004+2005/2005 B = 2005x(1/2+1/3+....+1/2004+1/2005) Vậy A = 2005

Đặt B = 2004+2003/2+2002/3+...+1/2004
B có 2004 phân số
tách số 2004 = 1+1+1+...+1(2004 số 1)
ghép 2004 số 1 vào từng nhóm như sau:
B=(1+ 2003/2)+ (1+ 2002/3)+...+(1+1/2004) +1
B = 2005/2+2005/3+......+2005/2004+2005/2005 
B = 2005x(1/2+1/3+....+1/2004+1/2005)
Vậy A = 2005