Question 1203648
<br>
Since the numbers are consecutive odd integers, the sequence is arithmetic, with a common difference of 2.<br>
The sum of the terms of an arithmetic sequence, in words (which is easier than a mathematical formula!) is<br>
(number of terms) times (average of all the terms)<br>
Since in an arithmetic sequence the average of all the terms is equal to the average of the first and last terms, the sum is<br>
(number of terms) times (average of first and last terms)<br>
We are given that the sum is a perfect 4th power of a positive integer; and we are given that the number of terms is 384.  So we need to have<br>
(384) * (average of first and last terms) = n^4<br>
((2^7)(3^1)) * (average of first and last terms) = n^4<br>
We want this perfect 4th power (and the resulting sum) to be as small as possible. It is clear that to get this the average of the first and last terms has to be (2^1)(3^3) = 54, making the sum of the terms<br>
(2^8)(3^4) = ((2^2)(3^1))^4 = 12^4 = 20736.<br>
ANSWER: b 20,736<br>
The problem doesn't ask us to find the terms of the sequence; but doing so is good further mathematical exercise.<br>
Letting the first term be a and the 384th term be (a+383(2)) = a+766, we need to have<br>
average of first and last terms = 54
{{{(a+a+766)/2=54}}}
{{{2a+766=108}}}
{{{2a=-658}}}
{{{a=-329}}}<br>
The first term of the sequence is -329 and the last term is -329+383(2) = -329+766 = 437.<br>