SOLUTION: The sum of the consecutive integers from -17 to x is 100. What is x?

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Question 984808: The sum of the consecutive integers from -17 to x is 100. What is x?
Found 2 solutions by Alan3354, Edwin McCravy:
Answer by Alan3354(69443) About Me  (Show Source):
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The sum of the consecutive integers from −17 to x is 100. What is x?
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The sum from -17 to +17 is zero for 35 integers.
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Now find integers starting at 18 whose sum is 100.
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18n+%2B+n%28n-1%29%2F2+=+100
n%5E2+%2B+35n+-+200+=+0
(n-5)*(n+40) = 0
n = 5
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40 integers total

Answer by Edwin McCravy(20055) About Me  (Show Source):
You can put this solution on YOUR website!
The sum of the consecutive integers from -17 to x is 100. What is x?

  an = a1 + (n-1)   
     x = -17 + (n-1)1     
     x = -17 + n - 1     
     x = n - 18
x + 18 = n

Sn = n%2F2[2a1 + (n-1)d]

100 = n%2F2[2(-17)+(n-1)1]

Multiply both sides by 2


200 = n[-34+(n-1)]

200 = n[-34+n-1]

200 = n(n-35)

Substitute x+18 for n

200 = (x+18)(x+18-35)

200 = (x+18)(x-17)

200 = x²+x-306

0 = x²+x-506

Since the leading coefficient is 1 and 
the middle coefficient is small compared to 506,
the numbers we need have to be near the square 
root of 506, which is about 22.49. So
the factoring numbers are near that:

0 = (x-22)(x+23) 

      x=22, x=-23.

x has to be positive so the answer is 22.


Edwin