First we figure out the pattern:
-4 + -1 + 8 + 23 + 44 + 71 + 104
start with -4
-4 + 3 = -1
-1 + 9 = 8
8 + 15 = 23
23 + 21 = 44
44 + 27 = 71
71 + 33 = 104
We are adding an odd multiple of 3 each time.
So we could write it this way:
start with -4
-4 + 3*1 = -1
-1 + 3*3 = 8
8 + 3*5 = 23
23 + 3*7 = 44
44 + 3*9 = 71
71 + 3*11 = 104
We start with this
-4 + -1 + 8 + 23 + 44 + 71 + 104
and go through a bunch of substitutions using the above list
-4 + -1 + 8 + 23 + 44 + 71 + 1*(71 + 3*11)
-4 + -1 + 8 + 23 + 44 + 71 + 71 + 1*3*11
-4 + -1 + 8 + 23 + 44 + 2*71 + 1*3*11
-4 + -1 + 8 + 23 + 44 + 2*(44 + 3*9) + 1*3*11
-4 + -1 + 8 + 23 + 44 + 2*44 + 2*3*9 + 1*3*11
-4 + -1 + 8 + 23 + 3*44 + 2*3*9 + 3*11
-4 + -1 + 8 + 23 + 3*(23 + 3*7) + 2*3*9 + 1*3*11
-4 + -1 + 8 + 23 + 3*23 + 3*3*7 + 2*3*9 + 1*3*11
-4 + -1 + 8 + 4*23 + 3*3*7 + 2*3*9 + 3*11
-4 + -1 + 8 + 4*(8 + 3*5) + 3*3*7 + 2*3*9 + 1*3*11
-4 + -1 + 8 + 4*8 + 4*3*5 + 3*3*7 + 2*3*9 + 1*3*11
-4 + -1 + 5*8 + 4*3*5 + 3*3*7 + 2*3*9 + 1*3*11
-4 + -1 + 5*(-1 + 3*3) + 4*3*5 + 3*3*7 + 2*3*9 + 1*3*11
-4 + -1 + 5*(-1) + 5*3*3 + 4*3*5 + 3*3*7 + 2*3*9 + 1*3*11
-4 + 6*(-1) + 5*3*3 + 4*3*5 + 3*3*7 + 2*3*9 + 1*3*11
-4 + 6*(-4 + 3*1) + 5*3*3 + 4*3*5 + 3*3*7 + 2*3*9 + 1*3*11
-4 + 6*(-4) + 6*3*1 + 5*3*3 + 4*3*5 + 3*3*7 + 2*3*9 + 1*3*11
7*(-4) + 6*3*1 + 5*3*3 + 4*3*5 + 3*3*7 + 2*3*9 + 1*3*11
-28 + 3*(6*1 + 5*3 + 4*5 + 3*7 + 2*9 + 1*11)
Using the formula for the nth term of an arithmetic sequence
The sequence 6,5,4,3,2,1 has kth term 6+(k-1)(-1) = 6-k+1 = 7-k
The sequence 1,3,5,7,9,11 has kth term 1+(k-1)(2) = 1+2k-2 = 2k-1
In summation form is
Edwin
