Question 621013: Use differences to find a pattern in the sequence.
-1, -2, -1, 5, 19, 44, 83
Assuming that the pattern continues, the eight term should be?
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website!
First Differences: -2 - (-1) = -1, -1 - (-2) = 1, 5 - (-1) = 6, 19 - 5 = 14, 44 - 19 = 25, 83 - 44 = 39
Second Differences: 1 - (-1) = 2, 6 - 1 = 5, 14 - 6 = 8, 25 - 14 = 11, 39 - 25 = 14
Third Differences: 5 - 2 = 3, 8 - 5 = 3, 11 - 8 = 3, 14 - 11 = 3
Third Differences are all the same, therefore we need a cubic polynomial.
If  , then the polynomial value is the value of the original first term, namely -1, so:
^3\ +\ b(1)^2\ +\ c(1)\ +\ d\ =\ -1)
which is to say:
Similarly:
Solve the 4X4 system of equations by the method of your choice then use the ordered quadruple solution to specify the coefficients of your cubic polynomial. Finally, calculate the value of the polynomial for 
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Or you can do it the easy way since you have the first 7 terms and only need the next one.
If Third Difference number 5 is the same as the others it is 3, then 14 plus 3 is 17 so 17 must be Second Difference number 6. Then 17 plus 39 is 56 which must be First Difference number 7, so term 7 which is 83 plus 56 is the 8th term, 139.
John
My calculator said it, I believe it, that settles it
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