Gọi `1 .2+2.3+3.4+4.5+5.6+...+2002.2003` là `A`
`A = 1 .2+2.3+3.4+4.5+5.6+...+2002.2003`
`=> 3A = 1.2.3 + 2.3.(4-1) + 3.4.(5-2)+4.5.(6-3) + 5.6.(7-4) + ... + 2002.2003 .(2004 - 2001)`
`=> 3A = 1.2.3 + 2.3.4 - 1.2.3 + 3.4.5 - 2.3.4 + 4.5.6 - 3.4.5 + 5.6.7 - 4.5.6 + ... + 2002 . 2003.2004 - 2001.2002.2003`
`=> 3A = 2002.2003.2004`
`=> A = \frac{2002.2003.2004}{3}`
`=> A = 2678684008`