SOLUTION: Construct a difference table to predict the next term of the sequence. −1, 3, 22, 61, 125, 219, 348, . . .

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Question 1163967: Construct a difference table to predict the next term of the sequence.
−1, 3, 22, 61, 125, 219, 348, . . .
Found 2 solutions by Boreal, greenestamps:
Answer by Boreal(15235)   (Show Source): You can put this solution on YOUR website!
from -1 to 3 is 4
to 22 is 19. the difference between 4 and 19 is 15.
then 39. next difference is 20. difference between 15 and 20 is 5. And all the others have difference 5
then 64. next difference is 25
then 94 next difference is 30
then 129 next difference is 35
then 169 (next difference is 40) and the term is 517
-1----
3----+4----
22---19---15----
61---39---20---5
125--64---25---5
219--94---30---5
348--129--35--5
517--169--40---5
Answer by greenestamps(13203)   (Show Source): You can put this solution on YOUR website!


Since the problem tells us to construct a difference table to predict the next term of the sequence, we can assume that in fact the sequence is generated by a polynomial function.
−1   3   22   61  125  219   348   original sequence
   4   19   39   64  94   129      first differences
    15    20   25  30   35         second differences
        5    5    5   5            constant third difference

There is the constant difference you need.  Continue that common difference for one more step:
−1   3   22   61  125  219   348
   4   19   39   64  94   129
    15    20   25  30   35
        5    5    5   5    5


Then work back up the array to find the next term of the sequence.
−1   3   22   61  125  219   348   517
   4   19   39   64  94   129   169
    15    20   25  30   35   40
        5    5    5   5    5


The next term in the sequence is 517.


 

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