document.write( "Question 1122929: A series of 384 consecutive odd integers has a sum that is a perfect fourth power of a positive integer. Find the smallest possible sum for this series. \n" ); document.write( "

Algebra.Com's Answer #739156 by greenestamps(13203) $\"\"$ $\"About$

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\n" ); document.write( "Let the first of the 384 consecutive odd numbers be x; then the last is x+383(2) = x+766.

\n" ); document.write( "The sum of 384 consecutive odd numbers with the first number x and the 384th number x+766 is

\n" ); document.write( " $\"%28384%29%28%28x%29%2B%28x%2B766%29%29%2F2+=+384%28x%2B383%29\"$

\n" ); document.write( "The objective is to find the smallest value of x for which that expression is the 4th power of a positive integer.

\n" ); document.write( "I don't know of an algebraic way to solve a problem like that. But either a spread sheet or a good graphing calculator can find the solution for you.

\n" ); document.write( "I used a TI83 with the function

\n" ); document.write( "(384(x+383))^(1/4)

\n" ); document.write( "and used the table feature to search for the solution, which is indicated by the function value being a whole number.

\n" ); document.write( "The smallest value of x that gives a whole number value for that function is 481; the function value is 24. That means the first odd number is 481 and the 384th odd number is 481+766 = 1247.

\n" ); document.write( "The question asks for the smallest sum of a series with a sum that is a 4th power of an integer.

\n" ); document.write( "Note since the whole number value of the function is 24, the sum of the series should be 24^4 = 331776.

\n" ); document.write( "And using the formula for the sum of the series, we in fact do have that sum:

\n" ); document.write( "384(481+383) = 331776. \n" ); document.write( "

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