Construct a difference table to predict the next term of the sequence. 1,5,15,31,53,81...

Question

Math homework question.

Answer 1

 

Number              Difference from previous                  Difference of Differences

   1

   5                                 4 

  15                              10                                                      6
  31                              16                                                      6
  53                              22                                                      6
  81                              28                                                      6

 

 

So, continuing this pattern, 

 115                             34                                                      6

Answer 2

The 2nd differences are constant = 6.

The next 1st difference in the table is 34.

The next term in the sequence is 115,

 

Please note that when the 2nd differences are constant, the first differences form an arithmetic sequence and the original sequence will be a quadratic.  In this case the first differences are given by 6n - 2 for integer n ≥ 1.  The original sequence is given by 3n² - 5n + 3 for integer n ≥ 1.