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?

##S_40 = 3220##
Use the sum formula :
##S_n = n/2 [ 2a + (n-1)d]##
and the ##n##-th term
##T_n = A +(n-1) d##
to solve for ##a## and ##d##.
This will get you
##S_12 = 186## and ##T_20 = 83##.
The equations are
##i) 12/2 [ 2a +11d ] = 186##
and
##ii) a + 19d = 83##
Solve like a linear system :
##{(12a + 66d = 186) (a + 19d = 83) :}##
##a = 83 – 19d##
sub in ##i)##
##12 (83 – 19d ) + 66d = 186##
##996 – 228d + 66d = 186##
##162 d = – 810##
##d = -5##
This means that
##a + 19 ( -5 ) = 83##
##a -95 = 83##
##a = 178##
Now
##S_40 = 20 [ 2(178) + 39 ( -5) ]##
## S_40 = 20 [ 356 – 195 ]##
##S_40 = 3220##