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\(E=-\dfrac{1}{3}\cdot\left(1+2+3\right)-\dfrac{1}{4}\left(1+2+3+4\right)-...-\dfrac{1}{50}\left(1+2+3+...+50\right)\)
\(=\dfrac{-1}{3}\cdot\dfrac{3\cdot4}{2}-\dfrac{1}{4}\cdot\dfrac{4\cdot5}{2}-...-\dfrac{1}{50}\cdot\dfrac{50\cdot51}{2}\)
\(=\dfrac{-4}{2}-\dfrac{5}{2}-...-\dfrac{51}{2}\)
\(=\dfrac{-\left(4+5+...+51\right)}{2}\)
\(=\dfrac{-\left(51+4\right)\cdot\dfrac{48}{2}}{2}=-\dfrac{1320}{2}=-660\)
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1/ S=1.2+2.3+3.4+...+50.51
=> 3S=1.2.3+2.3.3+3.4.3+...+50.51.3
=> 3S=1.2.3+2.3.(4-1)+3.4.(5-2)+...+50.51(52-49)
=> 3S=(1.2.3+2.3.4+3.4.5+...+50.51.52)-(1.2.3+2.3.4+...+49.50.51)
=> 3S=50.51.52 => S=50.51.52:3=44200
Đáp số: 44200
2/ A=12+22+32+42+...+502 = 1(2-1)+2(3-1)+3(4-1)+...+50(51-1)
=> A=(1.2+2.3+3.4+...+50.51)-(1+2+3+...+50)
=> A=S-\(\frac{50\left(50+1\right)}{2}\)=44200-1275
A=42925
Đáp số: 42925
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a, Ta có : S = 1*2 + 2*3 +3*4 + .... + 50*51
3S=1*2*3+2*3*3+3*4*3+....+50*51*3
3S=1*2*3+2*3*(4-1)+3*4*(5-2)+....+50*51*(52-49)
3S=1*2*3+2*3*4-1*2*3+3*4*5-2*3*4+...+50*51*52-49*50*51
3S=50*51*52
S=(50*51*52)/3=442000
b,Ta có 12 + 22 + 32 + ....... + n2=\(\frac{n\cdot\left(n+1\right)\cdot\left(2n+1\right)}{6}\)
=> 12 + 22 + 32 + ....... + 502= \(\frac{50\cdot\left(50+1\right)\cdot\left(2\cdot50+1\right)}{6}\)
=\(\frac{50\cdot51\cdot101}{6}\)= 42925
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\(\frac{3^2-1}{3^2}.\frac{4^2-1}{4^2}.\frac{5^2-1}{5^2}.....\frac{50^2-1}{50^2}\)
Tính biểu thức trên
![](https://rs.olm.vn/images/avt/0.png?1311)
\(=\left(1-\frac{1}{3^2}\right)\left(1-\frac{1}{4^2}\right)\left(1-\frac{1}{5^2}\right)...\left(1-\frac{1}{50^2}\right)\)
\(=\frac{8}{3\cdot3}\cdot\frac{15}{4\cdot4}\cdot\frac{24}{5\cdot5}\cdot....\cdot\frac{2499}{50\cdot50}\)
\(=\frac{\left(2\cdot4\right)\left(3\cdot5\right)\left(4\cdot6\right)...\left(49\cdot51\right)}{\left(3\cdot3\right)\left(4\cdot4\right)\left(5\cdot5\right)...\left(50\cdot50\right)}\)
\(=\frac{\left(2\cdot3\cdot4\cdot...\cdot49\right)\left(4\cdot5\cdot6\cdot...\cdot51\right)}{\left(3\cdot4\cdot5\cdot...\cdot50\right)\left(3\cdot4\cdot5\cdot...\cdot50\right)}\)
\(=\frac{2\cdot51}{50\cdot3}\)