Đặt biểu thức cần tính là A
Đặt B=1+22+32+42+...+1002=1+2(1+1)+3(2+1)+4(3+1)+...+100(99+1)
B=1+1.2+2+2.3+3+3.4+4+...+99.100+100=(1+2+3+4+...+100)+(1.2+2.3+3.4+...+99.100)
Đặt C=1.2+2.3+3.4+...+99.100 => 3.C=1.2.3+2.3.3+3.4.3+...+99.100.3=1.2.3+2.3.(4-1)+3.4.(5-2)+...+99.100.(101-98)
3.C=1.2.3-1.2.3+2.3.4-2.3.4-2.3.4+3.4.5-...-98.99.100+99.100.101=99.100.101 => C=33.100.101
Đặt \(D=1+2+3+4+...+100=\frac{100\left(1+100\right)}{2}=5050.\)
=> B=D+C=5050+33.100.101
A=(22+42+62++82+...+1002)-(1+32+52+72+...+992)
Đặt E=22+42+62+82+...+1002=22.(1+22+32+42+...+502)=22.[1+2.(1+1)+3(2+1)+4(3+1)+...+50(49+1)]
E=22.(1+1.2+2+2.3+3+3.4+4+...+49.50+50)=22.[(1+2+3+...+50)+(1.2+2.3+3.4+...+49.50] Tính tương tự như C và D
=> \(E=2^2.\left(\frac{50.\left(1+50\right)}{2}+\frac{49.50.51}{3}\right)=2^2.\left(1275+17.49.50\right)\)
Mặt khác ta có
B=(1+32+52+72+...+992)+(22+42+62+82+...+1002)=(1+32+52+72+...+992)+E => 1+32+52+72+...+992=B-E
=> A=E-(B-E)=2.E-B
\(\Rightarrow A=2^3\left(1275+17.49.50\right)-\left(5050+33.100.101\right)\)
