Răspuns:
Explicație pas cu pas:
a)
S = 1 - 2 + 3 - 4 + 5 - 6 + 7 - 8 + .........+ 101
S = ( 1-2) + (3-4)+(5-6)+(7-8) + .......(99 -100) +101
S = - 1 - 1 - 1 - 1 - 1......- 1 + 101
S = ( 100 : 2 ) × ( - 1) + 101
S = - 50 + 101
S = 51
Sau:
S = ( 1+3+5+7+.....+99+101) - (2+4+6+......+100)
1+3+5+....+101 = 51×(1+101):2 = 51×51
-> determin nr. termenilor= (101-1):2+1 = 51 termeni are suma
S = 51 × 51 - 2 ×(1+2+3+.....+50)
-> aplic formula sumei lui Gauss
S = 51 × 51 - 2 ×50 × (1+50):2
S = 51 × 51 - 50 × 51
S = 51 × (51 - 50)
S = 51 × 1
S = 51
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b)
S = - 1+2-3+4-5+6-7+8-.......- 201
S = (2+4+6+8+.....+200) - (1+3+5+.....+201)
S = 2×(1+2+3+4+....+100) - 101 ×101
S = 100×101 - 101 × 101
S = 101 × ( 100 - 101 )
S = - 101
1+3+5+7+....+201 =101 ×(201+1):2 = 101 ×101
-> determin cati termeni are suma cu ratia = 2
(201-1):2+1 = 200:2+1 = 101 termeni are suma
1+2+3+......+100 = 100 ×(1+100)/2 -> aplic formula sumei lui Gauss
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c)
S = - 1+2+3- 4+ 5 -6-7+8-.....+197
Grupand cate 4 termeni rezultatul este 0
(-1+2+3-4) +(5-6-7+8) +....... + 197
S = 0 + 197
S = 197
Sau grupand cate 2 termeni, obtin:
(-1+2) +(3-4) = 1 - 1 = 0
(5-6) +(-7+8) = - 1 + 1 = 0
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(193-194)+(-195+196) + 197 = 197
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d)
S = - 1 + 2 + 3-4+ 5-6- 7+8 - .....- 399
Grupand cate 2 termeni, obtin:
(-1+2) + (3-4) = 1 - 1 = 0
(5-6) + (-7+8) = - 1 + 1 = 0
(-9+10) +( 11 - 12) = 1 - 1 = 0
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399 : 4 = 99 rest 3
S = 1 - 399
S = - 398
