Question

1X2十2X3十3X4十4X5十···十98X99十99X100？

Answer 1

=1*（1+1）+2*（2+1）+3*（3+1）+4*（4+1）+...+98*（98+1）+99*（99+1）=1^2+1+2^2+2+3^2+3+4^2+4+...+98^2+98+99^2+99=(1^2+2^2+3^3+4^2+...+98^2+99^2)+(1+2+3+4+...+98+99)=99*(99+1)*(2*99+1)/6+(1+99)*99/2=(99*100*199)/6+100*99/2=(99*100*199)/6+100*99*3/6=(99*100*199+100*99)/6=99*100*(199+1)/6=99*100*200/6=33*50*200=33*10000=330000
99*100*200/6=33*50*200采用约分

Answer 2

1X2十2X3十3X4十4X5十···十98X99十99X100
=2X（1+3）+4X（3+5）+6X（5+7）+……+98X（97+99）+99X100
=8X(1X1+2X2+3X3+4X4+5X5+6X6+7X7+……+49X49)+99X100
=8X1/6X49X（49+1）X（2X49+1）+99X100
=8X1/6X49X50X99+99X100
=323400+9900
=333300

Answer 3

1X2十2X3十3X4十4X5十···十98X99十99X100
=2X（1+3）+4X（3+5）+6X（5+7）+……+98X（97+99）+99X100
=8X(1X1+2X2+3X3+4X4+5X5+6X6+7X7+……+49X49)+99X100
=8X1/6X49X（49+1）X（2X49+1）+99X100
=8X1/6X49X50X99+99X100
=323400+9900
=333300

Answer 4

an＝n(n+1)＝n^2+n
所以Sn＝n(n+1)(2n+1)/6+n(n+1)/2＝(n+1)n(2n+1+3)/6
＝n(n+1)(n+2)/3
当n＝99
S99＝999900/3＝333300