SOLUTION: I know there is a simpler way of doing this problem other than replacing the variable with each consecutive integer until they add up to 300, because I tried that and they didn't a

Question 177700: I know there is a simpler way of doing this problem other than replacing the variable with each consecutive integer until they add up to 300, because I tried that and they didn't add up. I would appreciate any help with this problem.
"The sum of the consecutive integers 1,2,3, ..., n is given by the formula . How many consecutive integers, starting with 1, must be added to get a sum of 300?"
Found 2 solutions by stanbon, gonzo:
Answer by stanbon(75887) (Show Source): You can put this solution on YOUR website!
S(n) = (1/2)*n(n+1). How many consecutive integers, starting with 1, must be added to get a sum of 300?
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Equation:
(n/2)(n+1) = 300
Multiply both sides by 2 to get:
n(n+1) = 600
n^2 +n - 600 = 0
(n-25)(n+24) = 0
Positive solution:
n = 25
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Cheers,
Stan H.
Answer by gonzo(654) (Show Source): You can put this solution on YOUR website!
you just set the equation to 300 and solve for n as follows:
n*(n+1)/2 = 300
(n^2 + n)/2 = 300
multiply both sides by 2 to get:
n^2 + n = 600
subtract 600 from both sides to get:
n^2 + n - 600 = 0
this is a quadratic equation.
factor to get:
(n+25) * (n-24) = 0
n = -25
or
n = 24
since n has to be positive, the only answer can be n = 24
substitute in original equation to get:
n * (n + 1)/2 = 300
which becomes:
(24*25)/2 = 300
which becomes:
600/2 = 300
which becomes:
300 = 300
equation is true so the value of n = 24 is good.
your answer is:
the number of consecutive integers, starting with 1, added together to get a sum of 300, is 24.
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