a,-53/24
b,chịu 

\(a,A=2^0+2^1+2^2+....+\)\(2^{2010}\)

\(\Rightarrow2A=2^1+2^2+2^3+....+2^{2011}\)

 \(2A-A=\left(2^1+2^2+2^3+...+2^{2011}\right)-\left(2^0+2^1+2^2+...+2^{2010}\right)\)

  \(A=2^{2011}-2^0\)

\(A=2^{2011}-1\)

\(b,B=1+3+3^2+...+3^{100}\)

\(\Rightarrow3B=3+3^2+3^3+...+3^{101}\)

\(3B-B=\left(3+3^2+3^3+...+3^{101}\right)-\left(1+3+3^2+...+3^{100}\right)\)

\(2B=3^{101}-1\)

\(\Rightarrow B=\frac{3^{101}-1}{2}\)

\(c,C=4+4^2+4^3+...+4^n\)

\(\Rightarrow4C=4^2+4^3+4^4+...+4^{n+1}\)

\(4C-C=\left(4^2+4^3+4^4+...+4^{n+1}\right)-\left(4+4^2+4^3+...+4^n\right)\)

\(3C=4^{n+1}-4\)

\(\Rightarrow C=\frac{4^{n+1}-4}{3}\)

\(d,D=1+5+5^2+...+5^{2000}\)

\(\Rightarrow5D=5+5^2+5^3+...+5^{2001}\)

\(5D-D=\left(5+5^2+5^3+...+5^{2001}\right)-\left(1+5+5^2+...+5^{2000}\right)\)

\(4D=5^{2001}-1\)

\(\Rightarrow D=\frac{5^{2001}-1}{4}\)

b)

B=1+3+3^2+3^3+..+3^100

=> 3B = 3 + 3^2 + 3^3 + ...+ 3^101

=> 3B - B = ( 3 + 3^2 + 3^3 + ...+ 3^101) - (1+3+3^2+3^3+..+3^100)

=> 2B = 3^101 - 1

=> B =( 3^101 - 1) / 2

A = (1-2).(1+2)+(3-4).(3+4)+(5-6).(5+6)+.....+(99-100).(99+100)

   = -1.3-1.7-1.11-......-1.199

   = -(3+7+11+....+99)

Trong dãy số 3;7;11;.....;99 có số số là : (99-3) : 4 + 1 = 25 (số)

=> A = -(3+99).25:2 = -1275

Tk mk nha

\(A=1^2-2^2+3^2-4^2+5^2-6^2+...+99^2-100^2\)

\(A=-3+\left(-7\right)+\left(-11\right)+...+\left(-199\right)\)

\(A=\frac{\left(-3+\left(-7\right)\right).50}{2}\)

\(A=-\frac{10.50}{2}\)

\(A=-250\)

Bài 2: 

a: \(\Leftrightarrow x=\dfrac{29}{60}\cdot\dfrac{-7}{5}=\dfrac{-203}{300}\)

b: \(\Leftrightarrow x\cdot\dfrac{2}{5}=\dfrac{29}{60}-\dfrac{3}{4}=\dfrac{29-45}{60}=\dfrac{-16}{60}=\dfrac{-8}{30}\)

\(\Leftrightarrow x=\dfrac{-8}{30}:\dfrac{2}{5}=\dfrac{-8\cdot5}{30\cdot2}=\dfrac{-40}{60}=-\dfrac{2}{3}\)