Question

suma lui gauss calculati: a=1+2+3+......+2022., b=86+87+88+......+149+150., c=2+4+6+8+.....+258+160, d=60+63+66+.....+282+285, e=3+7+11+.....+123+127, f=53+59+65+......+149+155.​

Answer 1

Explicație pas cu pas:

$a=1+2+3+......+2022 = \dfrac{2022 \cdot (2022 + 1)}{2} = \dfrac{2022 \cdot 2023}{2} = \bf 2045253$

$b=86+87+88+......+149+150 = (1 + 2 + 3 + ... + 149 + 150) - (1 + 2 + 3 + ... + 85) = \dfrac{150 \cdot (150 + 1)}{2} - \dfrac{85 \cdot (85 + 1)}{2} = \dfrac{150 \cdot 151}{2} - \dfrac{85 \cdot 86}{2} = 11325 - 3655 = \bf 7670$

$c=2+4+6+8+.....+258+260 = 2 \cdot (1 + 2 + 3 + ... + 129 + 130) = 2 \cdot \dfrac{130 \cdot (130 + 1)}{2} = 130 \cdot 131 = \bf 17030$

$d=60+63+66+.....+282+285 = 60 + (60 + 3) + (60 + 6) + ... + (60 + 222) + (60 + 225) = \underbrace{60 + 60 + ... + 60}_{76} + (3 + 6 + ... + 222 + 225) = 60 \cdot 76 + 3 \cdot (1 + 2 + ... + 74 + 75) = 4560 + 3 \cdot \dfrac{75 \cdot (75 + 1)}{2} = 4560 + 3 \cdot \dfrac{75 \cdot 76}{2} = 4560 + 8550 = \bf 13110$

$e=3+7+11+.....+123+127 = 3 + (3 + 4) + (3 + 8) + ... + (3 + 120) + (3 + 124) = \underbrace{3 + 3 + ... + 3}_{32} + (4 + 8 + ... + 120 + 124) = 3 \cdot 32 + 4 \cdot (1 + 2 + ... + 30 + 31) = 96 + 4 \cdot \dfrac{31 \cdot (31 + 1)}{2} = 96 + 2 \cdot 31 \cdot 32 = 96 + 1984 = \bf 2080$

$f=53+59+65+......+149+155 = 53 + (53 + 6) + (53 + 12) + ... + (53 + 96) + (53 + 102) = \underbrace{53 + 53 + ... + 53}_{18} + (6 + 12 + ... + 96 + 102) = 53 \cdot 18 + 6 \cdot (1 + 2 + ... + 16 + 17) = 954 + 6 \cdot \dfrac{17 \cdot (17 + 1)}{2} = 954 + 3 \cdot 17 \cdot 18 = 954 + 918 = \bf 1872$

Answer 2

Răspuns:

Explicație pas cu pas:

• a = 1+2+3+.......+2022

a = 2022×(1+2022) : 2

a = 1011 ×2023

a = 2 045 253

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• b = 86+87+88+.....+149+150

Primul mod:  Adaug și suma primele 85 de numere consecutive nenule pe care o scad apoi:

b = 1+2+3+.......+150 - (1+2+3+.......+85)

b = 150×(1+150) : 2 - 85×(1+85) : 2

b = 150×151 : 2 - 85 × 86 : 2

b = 22 650 : 2 - 7 310 : 2

b= 11 325 - 3 655

b = 7 670

• sau aflăm câte numere are suma:

150 - 86 + 1 = 65 termeni are suma

b = 65 × (86+150) : 2

b = 65 × 236 : 2

b = 65 × 118

b = 7670

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• c = 2+4=6+8+.......+258+260

- îl dau factor comun pe 2

c = 2×(1+2+3+4+.....+129+130)

c = 2×130×(1+130) : 2

c = 130×131

c = 17 030

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• d = 60+63+66+282+285

(285-60) : 3 + 1 = 75+1 = 76 termeni are suma

d = 76×(60+285) : 2

d = 13 110

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• e = 3+7+11+.....+123+127

(127-3) : 4 + 1 = 31 +1=32 termeni are suma

e = 32×(3+127) : 2

e = 32×130 : 2

e = 2080

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• f = 53+59+65+......+149+155

(155-53) : 6 + 1 = 17+1 = 18 termeni are suma

f = 18 × ( 53+155) : 2

f = 9 × 208

f = 1872