Question 453965: Use first and second differences to identify the next three terms in the following sequence: 12, 25, 44, 69, 100, 137, 180,...
answers:
a) 180, 229, 284
b) 180, 274, 335
c) 229, 284, 345
d) 229, 345, 412
Answer by solver91311(24713)   (Show Source):
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First differences:
13, 19, 25, 31, 37, 43
Second differences:
6, 6, 6, 6, 6
Since the second differences are all equal, the series can be modeled with a 2nd degree polynomial.
Consider the 2nd degree (quadratic) polynomial equation:
Where  is the ordinal of the term and  is the value of the term.
The value of the 1st term is  , therefore:
^2\ +\ b(1)\ +\ c\ =\ 12)
which is to say:
Similarly, using the 2nd and 3rd terms we can derive:
and
Solve the 3 by 3 system of linear equations using any desired method (I use the MDETERM function and Cramer's Rule in Excel to solve anything more complex than a simple 2X2) to calculate the coefficients of your quadratic model for your sequence. Then substitute 8, 9, and 10 for in the model to develop the 8th, 9th, and 10th terms of the sequence.
John
My calculator said it, I believe it, that settles it
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