<PRE><B>Question 1099657</B>
&lt;br&gt;
The numbers of numbers in the successive sets are
1, 2, 3, 4, ...&lt;br&gt;
Let&#39;s think about the process for finding the answer by answering a similar but much simpler problem: find the sum of the numbers in the set that has last number 19.&lt;br&gt;
We can see the answer in the statement of the problem; however, let&#39;s see how we can get the answer in a way that we can use to find the answer to the given problem.&lt;br&gt;
The &quot;first&quot; set contains 1 odd integer; the &quot;second&quot; contains 2; the third 3; and the fourth 4.&lt;br&gt;
1+2+3+4 = 10, and 19 is the 10th odd integer, so 19 is the last number in the 4th set.&lt;br&gt;
Then, since there are 4 numbers in that set, the four numbers in the set are 19, 17, 15, and 13.&lt;br&gt;
It is easy to see in this case that the sum of those numbers is 64.  But let&#39;s find that sum by a method that we can use for the given problem, where we can&#39;t see the whole set of numbers at once.&lt;br&gt;
We know at this point that 19 is the largest number in the 4th set.  The smallest number in the 4th set is 19 minus the common difference 2 three times, so the smallest number in the set is {{{19-3(2) = 19-6 = 13}}}.  Then the average of the numbers in the set is {{{(19+13)/2 = 16}}}; and with 4 numbers in the set, the sum is {{{16*4 = 64}}}.&lt;br&gt;
Now let&#39;s apply that same analysis to the given problem.&lt;br&gt;
The number 1259 is the 630th odd positive integer; and we are told that it is the last number in its set.&lt;br&gt;
So we need to find when 1+2+3+4+...+n is equal to 630.&lt;br&gt;
The sum of the integers from 1 to n is {{{(n(n+1))/2}}}; so we want to find a solution to
{{{(n(n+1))/2 = 630}}}
{{{n(n+1) = 1260}}}&lt;br&gt;
By one method or another, we find n=35.&lt;br&gt;
That means 1259 is the last number in the 35th set.&lt;br&gt;
Using the method we developed earlier, the first number in the 35th set is {{{1259-34*2 = 1259-68 = 1191}}}.&lt;br&gt;
Then the average of the numbers in the set is {{{(1259+1191)/2 = 2450/2 = 1225}}}.&lt;br&gt;
And then the sum of the numbers in that set is {{{35*1225 = 42875}}}.&lt;br&gt;
ANSWER:  The sum of the numbers in the set with last number 1259 is 42,875.
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