Question 1202468
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The response from the other tutor has at least a couple of significant errors, starting with a common difference of 12 instead of -12.<br>
Overlooking those errors, her presentation of the solution does little to help the student learn.<br>
The sequence consisting of the 16th through 53rd terms of an arithmetic sequence is itself an arithmetic sequence.<br>
The first term of that sequence (the 16th term of the original sequence) is the first term of the original sequence, plus the common difference 15 times: {{{543+15(-12)=543-180=363}}}.<br>
The last term of the new sequence ( the 53rd term of the original sequence) is the first term of the original sequence, plus the common difference 52 times: {{{543+52(-12)=543-624=-81}}}.<br>
The number of terms in the new sequence is {{{(53-16)+1=38}}}.<br>
The sum of the terms of the new sequence is the number of terms, times the average of the first and last terms:  {{{38((363-81)/2)=38*141=5358}}}.<br>
ANSWER: 5358<br>