Question 1172506
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There are dozens of similar ways to solve a problem like this.  Here is what I would do....<br>
Given:
Arithmetic sequence
7th term 17
13th term 12 more than the 7th term<br>
To find:
Sum of first 18 terms<br>
Solution....<br>
(1) The sum of the first 18 terms of an arithmetic sequence is 18 times the average of all the terms, which in a sequence with 18 terms is 18 times the average of the 9th and 10th terms (the two terms "in the middle").<br>
(2) The 13th term is the 7th term plus 13-7=6 times the common difference; since the 13th term is 12 more than the 7th, the common difference is 12/6 = 2.<br>
(3) So the 7th, 8th, 9th, and 10th terms of the sequence are 17, 19, 21, and 23.<br>
ANSWER: The sum of the first 18 terms is<br>
{{{18((21+23)/2) = 18(22) = (20-2)(20+2) = 20^2-2^2 = 400-4 = 396}}}<br>