6:

\(4D=2^2+2^4+...+2^{202}\)

=>3D=2^202-1

hay \(D=\dfrac{2^{202}-1}{3}\)

7: \(=\dfrac{1}{2}\left(\dfrac{2}{3\cdot5}+\dfrac{2}{5\cdot7}+...+\dfrac{2}{97\cdot99}\right)\)

\(=\dfrac{1}{2}\left(\dfrac{1}{3}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{7}+...+\dfrac{1}{97}-\dfrac{1}{99}\right)\)

\(=\dfrac{1}{2}\cdot\dfrac{32}{99}=\dfrac{16}{99}\)

\(P=\frac{\frac{1}{2003}+\frac{1}{2004}-\frac{1}{2005}}{\frac{5}{2003}+\frac{5}{2004}-\frac{5}{2005}}-\frac{\frac{2}{2002}+\frac{2}{2003}-\frac{2}{2004}}{\frac{3}{2002}+\frac{3}{2003}-\frac{3}{2004}}\)

\(P=\frac{\frac{1}{2003}+\frac{1}{2004}-\frac{1}{2005}}{5\left(\frac{1}{2003}+\frac{1}{2004}-\frac{1}{2005}\right)}-\frac{2\left(\frac{1}{2002}+\frac{1}{2003}-\frac{1}{2004}\right)}{3\left(\frac{1}{2002}+\frac{1}{2003}-\frac{1}{2004}\right)}\)

\(P=\frac{1}{5}-\frac{2}{3}=\frac{3-10}{15}=\frac{-7}{15}\)

a, \(\left(2-\dfrac{3}{2}\right)\left(2-\dfrac{4}{3}\right)\left(2-\dfrac{5}{4}\right)\left(2-\dfrac{6}{5}\right)\)

\(=\left(\dfrac{4}{2}-\dfrac{3}{2}\right)\left(\dfrac{6}{3}-\dfrac{4}{3}\right)\left(\dfrac{8}{4}-\dfrac{5}{4}\right)\left(\dfrac{10}{5}-\dfrac{6}{5}\right)\)

\(=\dfrac{1}{2}.\dfrac{2}{3}.\dfrac{3}{4}.\dfrac{4}{5}\)

\(=\dfrac{1}{5}\)

b. \(\dfrac{1}{2002}+\dfrac{2003.2001}{2002}-2003\)\(=\dfrac{1}{2002}+\dfrac{2003.2001}{2002}-\dfrac{2003.2002}{2002}\) = \(\dfrac{1+2003.2001-2003.2002}{2002}\) = \(\dfrac{1+\left(2003\left(2001-2002\right)\right)}{2002}\) = \(\dfrac{1+2003.\left(-1\right)}{2002}\) = \(\dfrac{1+\left(-2003\right)}{2002}\) = \(\dfrac{-2002}{2002}=-1\)

Chúc nguyễn hồng nhung học tốt

\(F=1-3+3^2-3^3+....+3^{2002}-3^{2003}+3^{2004}\)

\(3F=3-3^2+3^3-3^4+......+3^{2003}-3^{2004}+3^{2005}\)

\(3F+F=3^{2005}+1\)

\(\Leftrightarrow4F=3^{2005}+1\)

\(F=\frac{3^{2005}+1}{4}\)

Ta có: \(F=1-3+3^2-3^3+...+3^{2002}-3^{2003}+3^{2004}\)

\(\Rightarrow3F=3.\left(1-3+3^2-3^3+...+3^{2002}-3^{2003}+3^{2004}\right)\)

\(\Rightarrow3F=3-3^2+3^3-3^4+...+3^{2003}-3^{2004}+3^{2005}\)

\(\Rightarrow F+3F=1-3+3^2-3^3+...+3^{2002}-3^{2003}+3^{2004}+\)\(\left(3-3^2+3^3-3^4+...+3^{2003}-3^{2004}+3^{2005}\right)\)

\(\Rightarrow4F=1+3^{2005}\)

\(\Rightarrow F=\left(1+3^{2005}\right):4\)

c) 22/5 + 51/9 + 11/4 + 3/5 + 1/3 + 1/4
= 22/5 +3/5 +51/9 + 1/3 +11/4+1/4
= (22/5 +3/5) +(51/9 + 3/9) +(11/4+1/4)
= 25/5 +54/9 +12/4
= 5 +6 +3
= 14
d) (1/6 + 1/10 + 1/15) : (1/6 + 1/10 - 1/15) 
= (5/30 + 3/30 +2/30 ) :(5/30 +3/30 -2/30)
= 10/30 : 6/30
= 1/3 : 1/5
= 5/3

a) \(B=\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+...+\frac{1}{3^{2015}}\)

\(\Rightarrow3B=1+\frac{1}{3}+\frac{1}{3^2}+...+\frac{1}{3^{2014}}\)

\(\Rightarrow3B-B=1-\frac{1}{3^{2015}}\)

\(B=\frac{1-\frac{1}{3^{2015}}}{2}\)

giúp câu P luôn với bạn

a) \(\left(2-\frac{3}{2}\right)\left(2-\frac{4}{3}\right)\left(2-\frac{5}{4}\right)\left(2-\frac{6}{4}\right)\)

\(=\frac{1}{3}\left(-\frac{4}{3}+2\right)\left(-\frac{5}{4}+2\right)\left(-\frac{6}{4}+2\right)\)

\(=\frac{1}{2}.\frac{2}{3}\left(-\frac{5}{4}+2\right)\left(-\frac{6}{4}+2\right)\)

\(=\frac{1}{2}.\frac{2}{3}.\frac{3}{4}\left(-\frac{6}{4}+2\right)\)

\(=\frac{1.2.3\left(2-\frac{3}{2}\right)}{2.3.4}\)

\(=\frac{1.3\left(2-\frac{3}{2}\right)}{3.4}\)

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

\(=\frac{2-\frac{3}{4}}{4}\)

\(=\frac{1}{2.4}\)

\(=\frac{1}{8}\)

b) \(\left(\frac{2003}{2004}+\frac{2004}{2003}\right):\frac{8028025}{8028024}\)

\(=\frac{8028024\left(\frac{2003}{2004}+\frac{2004}{2003}\right)}{8028025}\)

\(=\frac{8028024.\frac{8028025}{4014012}}{8028025}\)

\(=\frac{16056050}{8028025}\)

= 2