Question

Calculaţi sumele:
a)S=+1 - 2+3 - 4+5 - 6+ 7-8+ ... +101;
b)S=-1 +2 - 3 + 4 - 5 + 6 - 7 + 8 - ... - 201
c)S=-1 + 2 + 3 - 4 + 5-6-7 + 8 - ... + 197;
d)S=-1+2+3 - 4+ 5-6-7 +8 - ... - 399.
Urgentttt!!! DAU COROANA!!!​

Answer 1

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