# 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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Noah G
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Mar 8, 2018

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We know that

Thus

#186= 12/2(2a + (11)d)#

#186 = 6(2a + 11d)#

#31 = 2a + 11d#

Now we know that

#83 = a + (20 - 1)d#

#83 = a + 19d#

We now have a system of equations:

#{(31 = 2a + 11d), (83 = a + 19d):}#

Substituting (2) into (1), we get

#31 = 2(83 - 19d) + 11d#

#31 = 166 - 38d + 11d#

#-135 = -27d#

#d = 5#

Now solving for

#83 - 19(5) = -12#

The sum is once again given by

#s_40 = 40/2(2(-12) + (39)5)#

#s_40 = 20(171)#

#s_40 = 3420#

Hopefully this helps!

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