Đặt S = 12 + 22 + 32 + ..... + 992 + 1002
<=> S = 1.1 + 2.2 + 3.3 + .... + 99.99 + 100.100
<=> S = 1.( 2 - 1 ) + 2.( 3 - 1 ) + 3.( 4 - 1 ) + ......... + 99.( 100 - 1 ) + 100.( 101 - 1 )
<=> S = 1.2 - 1 + 2.3 - 2 + 3.4 - 3 + ........ + 99.100 - 99 + 100.101 - 100
<=> S = ( 1.2 + 2.3 + 3.4 + ........ + 99.100 + 100.101 ) - ( 1 + 2 + 3 + ..... + 99 + 100 )
Đặt B = 1.2 + 2.3 + 3.4 + ..... + 99.100 + 100.101
C = 1 + 2 + 3 + .... + 99 + 100
=> S = B - C
B = 1.2 + 2.3 + 3.4 + .... + 99.100 + 100.101
<=> 3B = 1.2.3 + 2.3.3 + 3.4.3 + .... + 99.100.3 + 100.101.3
<=> 3B = 1.2.3 + 2.3.( 4 - 1 ) + 3.4. ( 5 - 2 ) + .... + 99.100.( 101 - 98 ) + 100.101.( 102 - 99 )
<=> 3B = 1.2.3 + 2.3.4 - 1.2.3 + 3.4.5 - 2.3.4 + ..... + 99.100.101 - 98.99.100 + 100.101.102 - 99.100.101
<=> 3B = 100.101.102
<=> B = ( 100.101.102 ) : 3
=> B = 343400
C = 1 + 2 + 3 + ... + 99 + 100
SSH của C là : ( 100 - 1 ) : 1 + 1 = 100 ( số hạng )
=> C = ( 100 + 1 ) . 100 : 2 = 5050
=> S = 343400 - 5050 = 338350
Vậy S = 338350
