SOLUTION: The smallest integer in a set of consecutive integers is -14. If the total sum of these integers is 31, then how many integers are in the set?

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Question 259678: The smallest integer in a set of consecutive integers is -14. If the total sum of these integers is 31, then how many integers are in the set?
Found 2 solutions by Fombitz, richwmiller:
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
The sum from -14 to -1 would be -%2814%29%2815%29%2F2=-105
So look for the integer such that n%28n%2B1%29%2F2=105%2B31=136
n%28n%2B1%29=272
n%5E2%2Bn-272=0
+%28n-16%29%28n%2B17%29=0
n-16=0
n=16
So the terms run from -14 to 16, for a total of 31 integers.

Answer by richwmiller(17219) About Me  (Show Source):
You can put this solution on YOUR website!
from -14 to 14 the total is 0
15+16=31
from -14 to 14 are 29 including 0
29+2=31
31 integers
the 16th would be 1
middle number * number of integers=total sum
1*31=31
ok

Another way
http://www.curiousmath.com/index.php?name=News&file=article&sid=95
middle number * number of integers=total sum
middle number * number of integers=31
factors of 31 are 1,31
31 must be the number of integers and 1 the middle number (16th)