Find a positive integral solution to the equation 1+3+5+… +(2n-1) 2+4+6+… +2n=(115)/(116).

1979 AMC 12 Problem 11

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11.

Find a positive integral solution to the equation

\frac{\substack{1+3+5+\cdots\\{}+(2n-1)}} {\substack{2+4+6+\cdots\\{}+2n}}=\frac{115}{116}.

110

115

116

231

The equation has no positive integral solutions.

Use the sums of the first

n

odd and first

n

even positive integers

The fraction simplifies to

\frac{n}{n+1}

The numerator is

n^2,

and the denominator is

n(n+1).

Thus

\frac{n}{n+1}=\frac{115}{116},

which gives

116n=115n+115,

n=115.

Therefore, the correct answer is B.