Question 1200343
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The problem says the n-th term is the sum of the first n terms of the sequence.  Since we count with positive numbers, n has to be positive.<br>
The numbers in the sequence are decreasing, so as we keep adding terms of this sequence, the sum will quickly become negative.<br>
Note also that, since the terms are all odd and the sequence is decreasing, in evaluating the sum of the terms to a particular point the later terms will cancel the earlier terms.<br>
It should be clear that when the term -7 is added, the sum of the terms of the sequence will be 0; so the last time the sum of the terms is positive is when the term -5 is added. That sum is<br>
 7 + 5 + 3 + 1 + (-1) + (-3) + (-5) = 7<br>
And that sum is the sum of the first 7 terms of the sequence, so the conditions of the problem are satisfied.  The sum of the first 7 terms of the sequence is 7, so<br>
ANSWER: 7<br>
For practice with the formal algebra for working with sums of arithmetic sequences, solving the problem using formal mathematics is a useful exercise.<br>
The sum of n terms of an arithmetic sequence is the number of terms, n, multiplied by the average of the terms.  And in an arithmetic sequence, the average of all the terms is the average of the first and last.<br>
Given first term a and common difference d, the n-th term of the sequence is a+(n-1)d.<br>
In this problem, a=7 and d=-2; so the n-th term is 7+(n-1)(-2) = 7-2n+2 = 9-2n.<br>
The sum of the first n terms (number of terms, times the average of the terms) is then<br>
{{{n((7+(9-2n))/2)}}}<br>
And we want that sum to be the number of the term:<br>
{{{n=n((7+(9-2n))/2)}}}<br>
{{{(7+(9-2n))/2=1}}}<br>
{{{16-2n=2}}}
{{{2n=14}}}
{{{n=7}}}<br>