SQUARE NUMBERS

Develop a spreadsheet in which you define the sequence of square numbers in terms of (a) the sequence of natural numbers, (b) the sequence of odd numbers, and (c) the sequence of triangular numbers. The headings and the first three rows of the spreadsheet are shown below.

			Square Numbers ((SQ)n)
Natural Numbers (Nn)	Odd Numbers (On)	Triangular Numbers (Tn)	Use natural numbers	Use odd numbers	Use triangular numbers
N1 =1 N2 =2 N3 =3	O1 =1 O2 =3 O3 =5	T1 =1 T2 =3 T3 =6	(SQ)1 =1 (SQ)2 =4 (SQ)3 =9	1 4 9	1 4 9

I don't really know how to describe how I found these formulas. I just made a longer list of the numbers side by side and looked at them and played with them until I found something that worked. I can verify the formulas, however, by substituting the O_n and T_n terms with their definitions in terms of n.

In all cases, (SQ)_n = n²

Natural Numbers

N_n = n

(SQ)_n = n² = (N_n)²

= n²

Odd Numbers

O_n = 2n -1

(SQ)_n = n² = [(O_n +1)/2]²

= [(2n -1 +1)/2]²

= [2n/2]²

= n²

Triangular Numbers

T_n = n(n+1)/2

OR(SQ)_n = n² = (T_n -T_n-1)²

= [n(n+1)/2 - (n-1)n/2]²

= [(n/2)(n+1 - n +1)]²

= [(n/2)(2)]²

= n²

(SQ)_n = n² = T_n + T_n-1

= n(n+1)/2 + (n-1)n/2

= (n/2)(n +1 +n -1)

= (n/2)(2n)

= n²

The second formula for triangular numbers was offered from the class.