MATH 2160

Date _____________                                                  Name _____________________________

Figurate Number

I.   The triangular numbers are the whole numbers that are represented by certain triangular arrays of dots.  The first five triangular numbers are shown.  Look for a pattern that the numbers follow.

 

 

 


 

  1                     3                           6                    10                        15

 ____                ____                    ____                 ____                      ____

1.  Complete the following table.

           

2.  Make a sketch to represent the sixth triangular number.

 

 

 

 

What is the sixth triangular number?  ________

3.  How many dots are there in the seventh triangular number? _________

Did you observe the following pattern?  ________          

        The 5th triangular number = The 4th triangular number + 5

        The 6th triangular number = The 5th triangular number + 6

4.  Complete the following equations.

The 7th triangular number = ______________________________________ + __________

The ______ triangular number = The 7th triangular number + ______ 

The ______ triangular number = The _______ triangular number + 9

According to the pattern, the nth triangular number = the (n 1)th triangular number + _____

 

Suppose T1 represents the 1st triangular number, T2 represents the 2nd triangular number, T3 represents the 3rd triangular number, and so on to Tn, which represents the nth triangular number.  The previous equations can be written as

 

T2 = T1 + 2

T3 = T2 + 3

T4 = T3 + 4

And so on.

 

The general statement would be: Tn = Tn-1 + n  (for n 2)

 

Note: If n = 10, then Tn represents the 10th triangular number and Tn-1 represents the 9th triangular number.

5.  Complete the following table.

 

T1 = 1 = 1                               T1 = 1 = 1                                                  T1 = (12 + 1)/2 = 1

T2 = 1 + 2 = 3                         T2 = 1 + 2 = 3                                           T2 = (22 + 2)/2 = 3

T3 = 3 + 3 = 6                         T3 = 1 + 2 + 3 = 6                                     T3 = (32 + 3)/2 = 6

T4 = 6 + 4 = 10                       T4 = 1 + 2 + 3 + 4 = 10                             T4 = (42 + 4)/2 = 10

T5 = 10 + 5 = ____                 T5 = 1 + 2 + 3 + 4 + 5 = ____                   T5 = (52 + 5)/2 = ____

T6 =  ____ + 6 = ____            T6 =                                     = ____           T6 = (____2 + ____)/2 = ____

T7 =  ____ + ____ = ____      T7 =                                           = ____     T7 = (____2 + ____)/2 = ____

T8 =  ____ + ____ = ____

T9 =  ____ + ____ = ____

T10 = ____ + ____ = ____

6.  Give two other general methods (formulas) for finding Tn:      Tn = _________________

                                                                                                            Tn = _________________

II.  The square numbers are the whole numbers that are represented by certain square arrays of dots.  The first five square numbers are shown.  Look for a pattern that the numbers follow.

 

 

 

 

  S1                    ____                    ____                       ____                             ____

1.  Complete the following table.

 

2.  Make a sketch to represent the sixth square number.

 

 

 

What is the sixth square number?  ________

3.  How many dots are there in the seventh square number? _________

4.  Complete the following equations.

 

The 7th square number = (____)2 or _____.

 

The _____ square number = 102 or _____.            Note: 102 = 10 x 10 = 100

 

According to the pattern, the nth square number = _____ squared or Sn = ______.

III.  The rectangular numbers are the whole numbers that are represented by certain rectangular arrays of dots.  The first five rectangular numbers are shown.  Look for a pattern that the numbers follow.

 

 

  

      R1                    ____                    ____                       ____                             ____

1.  Complete the following table.

 

 

2.  Make a sketch to represent the sixth rectangular number.

 

 

 

  

What is the sixth rectangular number?  ________

3.  How many dots are there in the seventh rectangular number? _________

Did you observe the following pattern?  ________

        The 5th rectangular number = 5 6 or 30.

        The 6th rectangular number = 6 7 or 42.

4.  Complete the following equations.

 

The 7th rectangular number = 7 _____ or _____.

 

The _____ rectangular number = 8 9 or _____.

 

The _____ rectangular number = _____ _____ or 110.

 

According to the pattern, the nth rectangular number = _____ _____.

 

5.  There is another way to look at Rn numbers that you have not explored yet.  Observe the following rectangular numbers:

 

             R1                    ____                    ____                       ____                             ____

6.  Complete the following table.

7.  According to the pattern, Rn  = _____ + _____.

Substitute the value you found for Sn into the formula you found above, then Rn  = _____ + _____.

8.  How is the last formula you obtained in #7 different from/the same as the last formula you obtained in #4 in this section?

__________________________________________________________________________ 

 

__________________________________________________________________________ 

9.  How are triangular, square, and rectangular numbers related to one another?

__________________________________________________________________________ 

__________________________________________________________________________ 

This page maintained by Dr. Peggy L. Moch

Last updated 10/24/2012 07:00 PM