What are the missing terms in the pattern? 1, 4, 12, 27, 55, 106, ___, ___

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embizze | High School Teacher | (Level 1) Educator Emeritus

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Given a finite number of terms of a sequence, there are an infinite number of possible sequences. Here is one possibility:

We are given the terms 1,4,12,27,55,106,...

Take the difference between successive terms. If these differences are all the same the terms could have been generated by a linear function; if they are different repeat the process. If the second differences are the same the sequence could be generated by a polynomial of degree 2. Continue this process. (Note: if the differences repeat, the generating function is exponential)

1      4      12      27      55      106
   \ /     \ /      \ /     \ /       \ /
    3      8        15     28      51
        \/      \/       \/       \/
        5       7       13      23
            \/      \/        \/
            2       6        10
                \/       \/
                4        4

Since the fouth order differences are the same, the generator could be a quartic polynomial. There are methods tofind the quartic -- Newton's method is one. Another is to realize that the polynomial will be `ax^4+bx^3+cx^2+dx+e` and use the given values to set up a system of equations. Easiest is to use a speadsheet or graphing utility with quartic regression capabilities:

The generator could be `1/6x^4-4/3x^3+19/3x^2-55/6x+5`

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The next two terms of this sequence would be 194,337

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