( - 1) + ( + 2 ) + ( - 3 ) + ( + 4 ) + ( - 5 ) + ( + 6 ) + ( - 7 ) + ( + 8 ) + ( - 9 ) + ( + 10 )= 5
= - 1 + 2 - 3 + 4 - 5 + 6 - 7 + 8 - 9 + 10 =
= 1 + 1 + 1 + 1 + 1 =
= 5
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sau: Observam ca numerele pare sunt pozitive si cele impare sunt negative.
( 2 + 4 + 6 + 8 + 10 ) - ( 1 + 3 + 5 + 7 + 9 ) = 5
= 30 - 25 =
= 5
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( - 1 ) + ( + 2 ) + ( - 3 ) + ( + 4 ) + ................. + ( - 99 ) + ( + 100 ) =
= ( 2 + 4 + 6 + ........ + 98 + 100 ) - ( 1 + 3 + 5 + ....... + 97 + 99 ) =
= 2 x ( 1 + 2 + 3 + ...... + 49 + 50 ) - [ 50 x ( 1 + 99 ) / 2 ] =
= 2 x ( 50 x 51 ) / 2 - ( 50 x 50 ) = sau 50 x ( 51 - 50 ) = 50 x 1 = 50
= 50 x 51 - 2 500 =
= 2 550 - 2 500 =
= 50
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1 + 3 + 5 + ....... + 97 + 99 =
→ stabilim cati termeni are suma; cu ratia 2 ( 3 - 1 = 2; 5 - 3 = 2 ....... 99 - 97 = 2)
( 99 - 1 ) : 2 + 1 = 98 : 2 + 1 = 49 + 1 = 50 termeni are suma
→ aplic formula sumei lui Gauss = 50 termeni x ( 1 + 99 ) / 2
