\(A=\dfrac{1}{2}+\dfrac{1}{2^2}+\dfrac{1}{2^3}+...+\dfrac{1}{2^{100}}\\ 2A=1+\dfrac{1}{2}+\dfrac{1}{2^2}+...+\dfrac{1}{2^{99}}\\ 2A-A=\left(1+\dfrac{1}{2}+\dfrac{1}{2^2}+...+\dfrac{1}{2^{99}}\right)-\left(\dfrac{1}{2}+\dfrac{1}{2^2}+\dfrac{1}{2^3}+...+\dfrac{1}{2^{100}}\right)\\ A=1-\dfrac{1}{2^{100}}\)
\(E=\dfrac{3^2}{2\cdot4}+\dfrac{3^2}{4\cdot6}+...+\dfrac{3^2}{198\cdot200}\\ =3^2\cdot\left(\dfrac{1}{2\cdot4}+\dfrac{1}{4\cdot6}+...+\dfrac{1}{198\cdot200}\right)\\ =9\cdot\dfrac{1}{2}\cdot\left(\dfrac{2}{2\cdot4}+\dfrac{2}{4\cdot6}+...+\dfrac{2}{198\cdot200}\right)\\ =\dfrac{9}{2}\cdot\left(\dfrac{1}{2}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{6}+...+\dfrac{1}{198}-\dfrac{1}{200}\right)\\ =\dfrac{9}{2}\cdot\left(\dfrac{1}{2}-\dfrac{1}{200}\right)\\ =\dfrac{9}{2}\cdot\dfrac{99}{200}\\ =\dfrac{891}{400}\)
\(B=1\cdot2+2\cdot3+3\cdot4+...+48\cdot49\\ 3B=1\cdot2\cdot3+2\cdot3\cdot3+3\cdot4\cdot3+...+48\cdot49\cdot3\\ 3B=1\cdot2\cdot\left(3-0\right)+2\cdot3\cdot\left(4-1\right)+3\cdot4\cdot\left(5-2\right)+...+48\cdot49\cdot\left(50-47\right)\\ 3B=1\cdot2\cdot3-0\cdot1\cdot2+2\cdot3\cdot4-1\cdot2\cdot3+3\cdot4\cdot5-2\cdot3\cdot4+...+48\cdot49\cdot50-47\cdot48\cdot49\\ 3B=-0\cdot1\cdot2+\left(1\cdot2\cdot3-1\cdot2\cdot3\right)+\left(2\cdot3\cdot4-2\cdot3\cdot4\right)+...+\left(47\cdot48\cdot49-47\cdot48\cdot49\right)+48\cdot49\cdot50\\ =0+48\cdot49\cdot50\\ =48\cdot49\cdot50\\ B=\dfrac{48\cdot49\cdot50}{3}\\ B=39200\)
C1:
\(C=1^2+2^2+3^2+...+48^2\\ =1^2+1+2^2+2+3^2+3+...+48^2+48-1-2-3-...-48\\ =1\cdot\left(1+1\right)+2\cdot\left(2+1\right)+3\cdot\left(3+1\right)+...+48\cdot\left(48+1\right)-\left(1+2+3+...+48\right)\\ =1\cdot2+2\cdot3+3\cdot4+...+48\cdot49-\left(1+2+3+...+48\right)\)
Gọi \(S=1\cdot2+2\cdot3+3\cdot4+...+48\cdot49\)
\(S=1\cdot2+2\cdot3+3\cdot4+...+48\cdot49\\ 3S=1\cdot2\cdot3+2\cdot3\cdot3+3\cdot4\cdot3+...+48\cdot49\cdot3\\ 3S=1\cdot2\cdot\left(3-0\right)+2\cdot3\cdot\left(4-1\right)+3\cdot4\cdot\left(5-2\right)+...+48\cdot49\cdot\left(50-47\right)\\ 3S=1\cdot2\cdot3-0\cdot1\cdot2+2\cdot3\cdot4-1\cdot2\cdot3+3\cdot4\cdot5-2\cdot3\cdot4+...+48\cdot49\cdot50-47\cdot48\cdot49\\ 3S=-0\cdot1\cdot2+\left(1\cdot2\cdot3-1\cdot2\cdot3\right)+\left(2\cdot3\cdot4-2\cdot3\cdot4\right)+\left(3\cdot4\cdot5-3\cdot4\cdot5\right)+...+\left(47\cdot48\cdot49-47\cdot48\cdot49\right)+48\cdot49\cdot50\\ 3S=0+48\cdot49\cdot50\\ 3S=48\cdot49\cdot50\\ S=\dfrac{48\cdot49\cdot50}{3}\\ S=39200\)
\(C=S-\left(1+2+3+...+48\right)\\ C=39200-\left(\dfrac{48\cdot49}{2}\right)\\ C=39200-1176\\ C=38024\)
