Bài giải :

$A=1.3+2.4+3.5+...+99.101$

$A=1.(2+1)+2.(3+1)+3.(4+1)+...+99.(100+1)$

$A=1.2+1+2.3+2+3.4+3+...+99.100+99$

$A=(1.2+2.3+3.4+...+99.100)+(1+2+3+...+99)$

Đặt $B=1.2+2.3+3.4+...+99.100$

$3B=1.2.3+2.3.3+3.4.3+...+99.100.3$

$3B=1.2.3+2.3.(4-1)+3.4.(5-2)+...+99.100.(101-98)$

$3B=1.2.3+2.3.4-1.2.3+3.4.5-2.3.4+...+99.100.101-98.99.100$

$3B=99.100.101$

$B=99.100.101:3$

$B=333300$

Đặt $C=1+2+3+...+99$

$C=\left( 99+1 \right).99:2=4950$

Vậy A = 333 300 + 4 950 =338 250