document.write( "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.
\n" ); document.write( "\"The sum of the consecutive integers 1,2,3, ..., n is given by the formula $\"%281%2F2%29%2An%28n%2B1%29\"$ . How many consecutive integers, starting with 1, must be added to get a sum of 300?\" \n" ); document.write( "

Algebra.Com's Answer #132703 by gonzo(654) $\"\"$ $\"About$

You can put this solution on YOUR website!
you just set the equation to 300 and solve for n as follows:
\n" ); document.write( "n*(n+1)/2 = 300
\n" ); document.write( "(n^2 + n)/2 = 300
\n" ); document.write( "multiply both sides by 2 to get:
\n" ); document.write( "n^2 + n = 600
\n" ); document.write( "subtract 600 from both sides to get:
\n" ); document.write( "n^2 + n - 600 = 0
\n" ); document.write( "this is a quadratic equation.
\n" ); document.write( "factor to get:
\n" ); document.write( "(n+25) * (n-24) = 0
\n" ); document.write( "n = -25
\n" ); document.write( "or
\n" ); document.write( "n = 24
\n" ); document.write( "since n has to be positive, the only answer can be n = 24
\n" ); document.write( "substitute in original equation to get:
\n" ); document.write( "n * (n + 1)/2 = 300
\n" ); document.write( "which becomes:
\n" ); document.write( "(24*25)/2 = 300
\n" ); document.write( "which becomes:
\n" ); document.write( "600/2 = 300
\n" ); document.write( "which becomes:
\n" ); document.write( "300 = 300
\n" ); document.write( "equation is true so the value of n = 24 is good.
\n" ); document.write( "your answer is:
\n" ); document.write( "the number of consecutive integers, starting with 1, added together to get a sum of 300, is 24.
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