a, \(S=\left(-2\right)+4+\left(-6\right)+8+...+\left(-198\right)+200\)
\(S=\left[\left(-2\right)+4\right]+\left[\left(-6\right)+8\right]+...+\left[\left(-198\right)+200\right]\)( 50 cặp )
\(S=2+2+...+2=2.50=100\)
Vậy S = 100
b, \(S=1+2-3-4+5+6-7-8+...-99-100+101+102\)
\(S=\left(1+2-3-4\right)+\left(5+6-7-8\right)+...+\left(97+98-99-100\right)+101+102\) ( 25 nhóm dư 2 )
\(S=-4+\left(-4\right)+...+\left(-4\right)+101+102\)
\(S=-4.25+101+102=-100+101+102=1+102=103\)
Vậy S = 103
c, \(S=1-2+3-4+...+1997-1998+1999\)
\(S=\left(1-2\right)+\left(3-4\right)+...+\left(1997-1998\right)+1999\)( 999 nhóm dư 1 )
\(S=-1+\left(-1\right)+...+\left(-1\right)+1999=-1.999+1999=-999+1999=1000\)
Vậy S = 1000
d, \(S=1-4+7-10+...-2998+3001\)
\(S=\left(1-4\right)+\left(7-10\right)+...+\left(2995-2998\right)+3001\)( 1500 Nhóm dư 1 )
\(S=-3+\left(-3\right)+...+\left(-3\right)+3001=-3.1500+3001=-4500+3001=-1499\)
Vậy S = -1499
e, \(S=1.2+2.3+3.4+...+19.20\)
\(\Rightarrow3S=1.2.3+2.3.3+3.4.3+...+19.20.3\)
\(\Rightarrow3S=1.2.3+2.3\left(4-1\right)+3.4.\left(5-2\right)+...+19.20.\left(21-18\right)\)
\(\Rightarrow3S=1.2.3+2.3.4-1.2.3+3.4.5-2.3.4+...+19.20.21-18.19.20\)
\(\Rightarrow3S=19.20.21=7980\Rightarrow S=7980\div3=2660\)
Vậy : S = 2660
