Question 1209385
<br>
Write the terms of the sequence in terms of the first term a and common difference d:<br>
a, a+d, a+2d, a+3d, ...<br>
The sum of n terms of the sequence is<br>
sum = (number of terms) * (average of all terms)<br>
And in an arithmetic sequence the average of all the terms is the average of the first and last terms:<br>
sum = (number of terms) * (average of first and last terms)<br>
For the sum of the first 10 terms, the 10th term is a+9d; the sum is<br>
{{{10((2a+9d)/2)}}}<br>
For the sum of the first 200 terms, the 200th term is a+199d; the sum is<br>
{{{200((2a+199d)/2)}}}<br>
The sum of the first 10 terms is 1, and the sum of the first 200 terms is 1/2:<br>
{{{10((2a+9d)/2)=1}}} --> {{{2a+9d=1/5}}}<br>
{{{200((2a+199d)/2)=1/2}}} --> {{{2a=199d=1/200}}}<br>
This is a straightforward problem of solving a pair of linear equations in two variables, but the numbers are ugly.  Use pencil and paper if you want; or solve the pair of equations using wolframalpha.com to find<br>
first term: 7951/76000
common difference: -39/38000<br>
Then use those to find<br>
ANSWER: the sum of the first 15 terms is {{{cross(1481/7600)}}} {{{4443/3040}}}<br>
The corrected solution was verified using an excel spreadsheet:<br>
first term: 7951/76000
common difference: -39/38000
sum of first 10 terms: 1
sum of first 15 terms: 4443/3040
sum of first 200 terms: 1/2<br>