<PRE><B>Question 1122929</B>
&lt;br&gt;
Let the first of the 384 consecutive odd numbers be x; then the last is x+383(2) = x+766.&lt;br&gt;
The sum of 384 consecutive odd numbers with the first number x and the 384th number x+766 is&lt;br&gt;
{{{(384)((x)+(x+766))/2 = 384(x+383)}}}&lt;br&gt;
The objective is to find the smallest value of x for which that expression is the 4th power of a positive integer.&lt;br&gt;
I don&#39;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.&lt;br&gt;
I used a TI83 with the function&lt;br&gt;
(384(x+383))^(1/4)&lt;br&gt;
and used the table feature to search for the solution, which is indicated by the function value being a whole number.&lt;br&gt;
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.&lt;br&gt;
The question asks for the smallest sum of a series with a sum that is a 4th power of an integer.&lt;br&gt;
Note since the whole number value of the function is 24, the sum of the series should be 24^4 = 331776.&lt;br&gt;
And using the formula for the sum of the series, we in fact do have that sum:&lt;br&gt;
384(481+383) = 331776.</PRE>