Question

Cara menghitung
1) 1-2+3-4+5-6+7-8+...-100
2) -100-99-98-...-2-1-0+1+2+...+48+49+50

Answer 1

*1)

#Deret Aritmatika
1-2+3-4+5-6...-100
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[CARA -1]
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Sn = 1-2+3-4+5-6...-100
Sn = (1+3+5..+99) - (2+4+6..+100)
Sn = Sn.Deret 1 - Sn.Deret 2

#Deret 1
Sn = 1+3+5..+99
Sn = U1+U2+U3..+Un

a = U1 → 1
b = U2 - U1 → 2

Un = a + (n-1)b
99 = 1 + (n-1)2
n = 100/2
n = 50

Sn = (n/2)(a+Un)
S50 = (50/2)(1+99)
S50 = 25(100)
S50 = 2500
Sn.Deret 1 = 2500

#Deret 2
Sn = 2+4+6...+100

a = 2
b = 2

Un = a + (n-1)b
100 = 2 +(n-1)2
n = 100/2
n = 50

Sn = (n/2)(a+Un)
S50 = (50/2)(2+100)
S50 = 25(102)
S50 = 2550
Sn.Deret 2 = 2550

Sn = Sn.Deret 1 - Sn.Deret 2
Sn = 2500 - 2550
Sn = -50
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[CARA -2]
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• 1-2 + 2-4 + 5-6 + 7-8 + 9-10....-100
• -1 + -1 + -1 + -1 + -1...-100
• -5 ... -100
• (1~10)+(11~20)...+(91~100)
= -5 × (100/10)
= -5 × 10
= -50

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Kesimpulan :
" 1-2+3-4...+99-100 = -50 "
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*2)

#Deret Aritmatika
-100-99...+50
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Sn = -100-99...-1-0+1... +50

nilai positif hanya pada interval 1-50, sedangkan negatif pada interbal 1-100, maka : ↓

= -100-99..-1-0+1..+50
= -100-99..-51 + (-50...-1-0+1...+50)
= -100-99...-51
= - (100+99..+51)
= - Sn
a = 100
b = -1

Un = a + (n-1)b
51 = 100 +(n-1)-1
-49 = -n +1
n = 49+1
n = 50

-Sn = (50/2)(100+51)
-Sn = 25(151)
-Sn = 3775
Sn = 3775/-1
Sn = -3775

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Kesimpulan :
" -100-99...+49+50 = -3775 "
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