**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 5^{th} triangular number = The 4^{th}
triangular number + 5

·
The 6^{th} triangular number = The 5^{th}
triangular number + 6

4. Complete the following equations.

The 7^{th} triangular
number = ______________________________________ + __________

The ______ triangular number =
The 7^{th} triangular number + ______

The ______ triangular number =
The _______ triangular number + 9

According to the pattern, the **
nth triangular number** **= the (n – 1)**^{th} triangular number +
_____

Suppose T_{1} represents the 1^{st}
triangular number, T_{2} represents the 2^{nd} triangular
number, T_{3} represents the 3^{rd} triangular number, and so on
to T_{n}, which represents the nth triangular number. The previous
equations can be written as

T_{2} = T_{1} + 2

T_{3} = T_{2} + 3

T_{4} = T_{3} + 4

And so on.

The general statement would be: **T**_{n}
= T_{n-1} + n (for n ³ 2)

Note: If n = 10, then T_{n}
represents the 10^{th} triangular number and T_{n-1} represents
the 9^{th} triangular number.

5. Complete the following table.

T_{1} = 1 = 1 T_{1} = 1 =
1 T_{1} = (1^{2}
+ 1)/2 = 1

T_{2} = 1 + 2 = 3 T_{2} = 1 + 2 =
3 T_{2} = (2^{2} +
2)/2 = 3

T_{3} = 3 + 3 = 6 T_{3} = 1 + 2 + 3 =
6 T_{3} = (3^{2} + 3)/2 = 6

T_{4} = 6 + 4 = 10 T_{4} = 1 + 2 + 3 + 4 =
10 T_{4} = (4^{2} + 4)/2 = 10

T_{5} = 10 + 5 = ____ T_{5} = 1 + 2 + 3 + 4 + 5
= ____ T_{5} = (5^{2} + 5)/2 = ____

T_{6} = ____ + 6 = ____ T_{6} =
= ____ T_{6} = (____^{2}
+ ____)/2 = ____

T_{7} = ____ + ____ = ____ T_{7} =
= ____ T_{7} = (____^{2}
+ ____)/2 = ____

T_{8} = ____
+ ____ = ____

T_{9} = ____
+ ____ = ____

T_{10} = ____
+ ____ = ____

6. Give two
other general methods (formulas) for finding T_{n}: T_{n} =
_________________

T_{n} = _________________

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.

S_{1 } ____
____ ____ ____

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 7^{th} square number =
(____)^{2} or _____.

The _____ square number = 10^{2}
or _____. Note: 10^{2} = 10 x 10 = 100

According to the pattern, the **nth
square number = _____ squared or S**_{n} = ______.

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.

R_{1 }
____ ____ ____
____

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 5^{th} rectangular number = 5
× 6 or 30.

·
The 6^{th} rectangular number = 6
× 7 or 42.

4. Complete the
following equations.

The 7^{th} 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 R_{n} numbers that you have not explored yet.
Observe the following rectangular numbers:

R_{1
} ____ ____
____ ____

6. Complete the following table.

7. According to the pattern, **
R**_{n } = _____ + _____.

Substitute the value you found for
S_{n} into the formula you found above, then **R**_{n } = _____ +
_____.

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?

__________________________________________________________________________

__________________________________________________________________________