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Zorluk: OrtaConsecutive Integers and Number Sets

Set SS consists of consecutive integers. The sum of all positive integers in set SS is 105105, and the sum of all integers in set SS is −31-31. How many negative integers are contained in set SS?

Cevap: 16

Cevap

There are 16 negative integers in set SS.
The correct response is 16. By setting up the sum of consecutive positive integers starting at 1, we determine that the set contains positive integers up to 14, which sum to 105. Subtracting 105 from the total set sum of -31 reveals that the negative integers must sum to -136. The consecutive negative integers -1, -2, ..., -p sum to -136 when p = 16, since 16 × 17 / 2 = 136.

Adım Adım Çözüm

1
Find the maximum positive integer kk in set SS
The largest positive integer in set SS is 1414
Because set SS consists of consecutive integers, the positive integers are 1,2,…,k1, 2, \dots, k. The sum formula k(k+1)2=105\frac{k(k+1)}{2} = 105 leads to k(k+1)=210k(k+1) = 210. Factoring 210210 into two consecutive integers gives 14×1514 \times 15, so k=14k = 14.
2
Calculate the sum of all negative integers in set SS
The sum of all negative integers is −136-136
The total sum of set SS is the sum of its negative integers plus 00 plus the sum of its positive integers: −31=Sneg+0+105  ⟹  Sneg=−136-31 = S_{\text{neg}} + 0 + 105 \implies S_{\text{neg}} = -136.
3
Determine the count pp of negative integers
The number of negative integers is 1616
The negative integers are −1,−2,…,−p-1, -2, \dots, -p. Their sum is −p(p+1)2=−136  ⟹  p(p+1)=272-\frac{p(p+1)}{2} = -136 \implies p(p+1) = 272. Solving p(p+1)=272p(p+1) = 272 gives p=16p = 16 because 16×17=27216 \times 17 = 272.

Anahtar Kavram

Consecutive integer set properties and partitioning sets into positive and negative components using arithmetic series formulas.
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Konular

Consecutive Integers and Number Sets

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