Patterns in a Whole Number Array


An array of  Whole numbers is created by placing rows as follows:

     1 in the first row,

     2,  3,  4  in the second row

     5, 6,  7,  8,  9  in the third row and so forth.

What patterns can be found in the array?

  a.  What is the first term of the  nth row?  the last term of the nth row?

  b.  What is the sum of the terms in the nth row?   HINT?   PROOF?

  c.  What is the sum of ALL the terms in the first   n   rows (in terms of n)?

  d.  What is the total number of terms in the first n rows?

  e.  "Diagonals" are parallel to an edge.  For Example,  4, 7, 12, 19, . . .    is a diagonal.   

            What are its next two terms of this diagonal?

            What is its term in the   nth    row for this diagonal?   
                   Note:  This is not the  nth  term of the diagonal.

 f.  Examine the sequences in the columns.  

            For example, center column is 1, 3, 7, 13, 21, . . .    Next two terms?

                    What is the center column term in the  nth  row? 

 g.  In what way are all of the diagonals "alike"  except for the starting terms?

          Hint:    Write the first few terms.   Then write a row of differences of consecutive terms.  Repeat if possible.

          See:    Finite Differences

  h.  Can you generate this array, possibly skewed to a right triangle shape, with a spreadsheet?

Editorial Comment:   In answering the above questions,   "how" you arrived at the answer may be more important than the answer itself.

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