SOLUTION: The sum of the first 12 terms of an arithmetic series is 186, and the 20th term is 83. What is the sum of the first 40 terms?

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Question 501418: The sum of the first 12 terms of an arithmetic series is 186, and the 20th term is 83. What is the sum of the first 40 terms?
Answer by Edwin McCravy(8924) About Me  (Show Source):
You can put this solution on YOUR website!
The sum of the first 12 terms of an arithmetic series is 186,
S%5Bn%5D=expr%28n%2F2%29%282a%5B1%5D%2B%28n-1%29d%29

S%5B12%5D=expr%2812%2F2%29%282a%5B1%5D%2B%2812-1%29d%29

186=6%282a%5B1%5D%2B11d%29

186%2F6=expr%286%2F6%29%282a%5B1%5D%2B11d%29

31=1%282a%5B1%5D%2B11d%29

31=2a%5B1%5D%2B11d

2a%5B1%5D%2B11d=31

and the 20th term is 83.
a%5Bn%5D=a%5B1%5D%2B%28n-1%29d
a%5B20%5D=a%5B1%5D%2B%2820-1%29d
83=a%5B1%5D%2B19d
a%5B1%5D%2B19d=83
So we have this system:
system%282a%5B1%5D%2B11d=31%2C+a%5B1%5D%2B19d=83%29
Solve that system and get a%5B1%5D=-12, d=5
To get the sum of the first 40 terms:
S%5Bn%5D=expr%28n%2F2%29%282a%5B1%5D%2B%28n-1%29d%29
S%5B40%5D=expr%2840%2F2%29%282%28-12%29%2B%2840-1%295%29
S%5B40%5D=20%28-24%2B%2839%295%29
S%5B40%5D=20%28-24%2B195%29
S%5B40%5D=20%28171%29
S%5B40%5D=20%28171%29
S%5B40%5D=3420
Edwin