\(\left(2^{78}+2^{79}+2^{80}\right):\left(2^{77}+2^{76}+2^{75}\right)=\left(2^{78}.7\right):\left(2^{75}.7\right)=2^{78}:2^{75}=8\)

Ta có: \(M-N=77^2+75^2+....+1^2-\left(76^2+74^2+...+2^2\right)\)

\(=77^2+75^2+....+1^2-76^2-74^2-...-2^2\)

\(=\left(77^2-76^2\right)+\left(75^2-74^2\right)+...+\left(3^2-2^2\right)+1^2\)

\(=\left(77-76\right)\left(77+76\right)+\left(75-74\right)\left(75+74\right)+...+\left(3-2\right)\left(3+2\right)+1\)

\(=77+76+75+74+...+3+2+1\)

\(=\frac{\left[\left(77-1\right):1+1\right].\left(1+77\right)}{2}=\frac{77.78}{2}=3003\)

Thay vào S, ta có: \(S=\frac{M-N-3}{3000}=\frac{3003-3}{3000}=\frac{3000}{3000}=1\)

\(\dfrac{2}{7\times9}+\dfrac{2}{9\times11}+\dfrac{2}{11\times13}+...+\dfrac{2}{75\times77}\)
\(=\dfrac{1}{7}-\dfrac{1}{9}+\dfrac{1}{9}-\dfrac{1}{11}+\dfrac{1}{11}-\dfrac{1}{13}+...+\dfrac{1}{75}-\dfrac{1}{77}\)
\(=\dfrac{1}{7}-\dfrac{1}{77}\)
\(=\dfrac{11}{77}-\dfrac{1}{77}\)
\(=\dfrac{10}{77}\)

\(\dfrac{2}{7x9}+\dfrac{2}{9x11}+\dfrac{2}{11x13}+...+\dfrac{2}{75x77}\)

= \(\dfrac{9-7}{7x9}+\dfrac{11-9}{9x11}+\dfrac{13-11}{11x13}+...+\dfrac{77-75}{75x77}\)

= \(\dfrac{9}{7x9}-\dfrac{7}{7x9}+\dfrac{11}{9x11}-\dfrac{9}{9x11}+\dfrac{13}{11x13}-\dfrac{11}{11x13}+...+\dfrac{77}{75x77}-\dfrac{75}{75x77}\)

=\(\dfrac{1}{7}-\dfrac{1}{9}+\dfrac{1}{9}-\dfrac{1}{11}+\dfrac{1}{11}-\dfrac{1}{13}+...+\dfrac{1}{75}-\dfrac{1}{77}\)

=\(\dfrac{1}{7}-\dfrac{1}{77}=\dfrac{11}{77}-\dfrac{1}{77}=\dfrac{10}{77}\)

=(2^78+79+80):(2^77+76+75)

=2²³⁷:2²²⁸

=2^237-228

=2⁹

=512

​khó quá 

M-N=(77^2-76^2)+(75^2-74^2)+...+(3^2-2^2)+1(dùng HĐT số 3)

M-N=1+2+3+4+5+...+76+77

M-N=3003

Nên (M-N-3)/3=3000/3=1000

Nhớ tich cho mình nha bạn

\(\frac{2^{78}\cdot\left(1+2+2^2\right)}{2^{75}\cdot\left(1+2+2^2\right)}=2^3=8\)

\(a;\frac{2^{78}+2^{79}+2^{80}}{2^{77}+2^{76}+2^{75}}=\frac{2^{78}\left(1+2+2^2\right)}{2^{75}\left(1+2+2^2\right)}=2^3=8\)

b) \(\frac{5^{56}+5^7}{5^{49}+1}=\frac{5^7\left(5^{49}+1\right)}{5^{49}+1}=5^7\)