**Introduction**

Learning number patterns is crucial since it supports the development of a solid mathematical foundation and facilitates dealing with numbers. We will be able to understand the connections between different numbers. Thinking abilities will be improved, including observing sequences and predicting what will happen next.

This article will discuss polygonal numbers that incorporate number patterns, how to describe a sequence of polygonal numbers using dots, and how to determine a given polygonal number depending on its order.

**What are polygonal numbers?**

**Definition**

In mathematics, polygonal numbers are nonnegative integers that are represented by regular polygons made of geometrically arranged, evenly spaced points. When the number of equidistant points used to represent a polygon expands, it enlarges in size in a predictable way.

Typically, a geometric figure made up of dots is used to represent polygonal numbers. The number of dots used rises in subsequent polygons, expanding from a central point outward. Triangles, squares, pentagons, and hexagons are the shapes of the first four polygonal numbers.

The foundation of polygonal numbers, as demonstrated in the image above, is to consider all polygonal forms and sizes as numerical values. The blue dots represent triangular numbers that fall into the sequence of 1, 3, 6, 10, and so forth. The green dots represent square numbers, which lead to the numbers 1, 4, 9, 16, 25, and so on. The orange dots represent pentagonal numbers in the following order: 1, 5, 12, 22, 35, 51, 70, etc. The sequence of hexagonal numbers, represented by the red dots, is 1, 6, 15, 28, 45, and so forth.

**How to find the next polygonal number?**

Let us use dots to show the next triangular, square, and pentagonal numbers.

The augmenting array of dots of polygonal numbers has the following stages.

The first polygonal number is always 1.

The polygon’s vertex count is the second number.

The third and subsequent values can be obtained by adding one point to each of two adjacent sides, followed by creating another regular polygon on top of the first.

**Triangular Numbers**

The first triangular number is 1.

The second triangular number is 3 since a triangle has three vertices.

For the third triangular number, we will add one point to each of the two adjacent sides. Equal distances must separate the points. Having three points on each side of a regular triangle, we can now draw another triangle. The third triangular number shows the total number of points.

Hence, the third triangular number is 6.

**Square Numbers**

One is the first square number.

Since a square has four vertices, the second square number is 4.

For the third square number, we will add one point to each of the two edges. The points must be separated by equal distances. Now that a regular square has three points on each side, we can design another square. The total number of points shows the third square number.

So, 9 is the third square number.

**Pentagonal Numbers**

One is the first square number.

A pentagon has five vertices; hence 5 is the second square number.

Add one point to each of the two sides to create the third pentagonal number. The points must be separated by identically sized gaps. We may now draw a second pentagon with three points on each side. The total points give the third pentagonal number.

Therefore, 12 is the third pentagonal number.

**List of Polygonal Numbers**

The following numbers show some of the few triangular, square, pentagonal, and hexagonal numbers.

*These are the initial triangular numbers:*

1, | 3, | 6, | 10, | 15, | 21, | 28, | 36, | 45, | 55, | 66, | 78, | 91, |

105, | 120, | 136, | 153, | 171, | 190, | 210, | 231, | 253, | 276, | 300, | 325, | 351, |

378, | 406, | 435, | 465, | 496, | 528, | 561, | 595, | 630, | 666, | 703, | 741, | 780, |

820, | 861, | 903, | 946, | 990, | 1035, | 1081, | 1128, | 1176, | 1225, | 1275, | 1326, | 1378, |

1431, | 1485, | 1540, | 1596, | 1653, | 1711, | 1770, | 1830, | 1891, | 1953, | 2016, | … |

*These are the initial square numbers:*

1, | 4, | 9, | 16, | 25, | 36, | 49, | 64, | 81, | 100, | 121, | 144, | 169, |

196, | 225, | 256, | 289, | 324, | 361, | 400, | 441, | 484, | 529, | 576, | 625, | 676, |

729, | 784, | 841, | 900, | 961, | 1024, | 1089, | 1156, | 1225, | 1296, | 1369, | 1444, | 1521, |

1600, | 1681, | 1764, | 1849, | 1936, | 2025, | 2116, | 2209, | 2304, | 2401, | 2500, | 2601, | … |

*These are the initial pentagonal numbers:*

1, | 5, | 12, | 22, | 35, | 51, | 70, | 92, | 117, | 145, | 176, | 210, | 247, |

287, | 330, | 376, | 425, | 477, | 532, | 590, | 651, | 715, | 782, | 852, | 925, | 1001, |

1080, | 1162, | 1247, | 1335, | 1426, | 1520, | 1617, | 1717, | 1820, | 1926, | 2035, | 2147, | 2262, |

2380, | 2501, | 2625, | … |

*These are the initial hexagonal numbers:*

1, | 6, | 15, | 28, | 45, | 66, | 91, | 120, | 153, | 190, | 231, | 276, | 325, |

378, | 435, | 496, | 561, | 630, | 703, | 780, | 861, | 946, | 1035, | 1128, | 1225, | 1326, |

1431, | 1540, | 1653, | 1770, | 1891, | 2016, | 2145, | 2278, | 2415, | 2556, | 2701, | 2850, | 3003, |

3160, | 3321, | 3486, | … |

**The Formula for Finding the n**^{th}** Polygonal Number**

^{th}

For any natural number n, here are the formulas for triangular, square, pentagonal, and hexagonal numbers.

*Triangular Numbers*

T_{n}=$\frac{n ( n+1 )}{2}$

*Square Numbers*

S_{n}=n^{2}

*Pentagonal Numbers*

P_{n} = $\frac{( 3n^2-n )}{2}$

*Hexagonal Numbers*

H_{n} = 2n^{2} – n

The table below shows the formula’s first four numbers in each sequence.

Polygon | Formula | n=1 | n=2 | n=3 | n=4 |

Triangular | T_{n}=$\frac{n ( n+1 )}{2}$ | T_{1}=$\frac{1 ( 1+1 )}{2}$T _{1}=1 | T_{1}=$\frac{2 ( 2+1 )}{2}$T _{1}=3 | T_{1}=$\frac{3 ( 3+1 )}{2}$T _{1}=6 | T_{1}=$\frac{4 ( 4+1 )}{2}$T _{1}=10 |

Square | S_{n} = n^{2} | S_{1} = 1^{2}S _{1}=1 | S_{2}=2^{2}S _{2}=4 | S_{3}=3^{2}S _{3}=9 | S_{4}=4^{2}S _{4}=16 |

Pentagonal | P_{n} = $\frac{( 3n^2-n )}{2}$ | P_{1} = $\frac{( 3(1)^2-1 )}{2}$P _{1}=1 | P_{2}= $\frac{( 3(2)^2-2 )}{2}$P _{2}=5 | P_{3}= $\frac{( 3(3)^2-3 )}{2}$P _{3}=12 | P_{4}= $\frac{( 3(4)^2-4 )}{2}$P _{4}=22 |

Hexagonal | H_{n} = 2n^{2} – n | H_{1}=2(1)^{2}-1H _{1}=1 | H_{2}=2(2)^{2}-2H _{2}=6 | H_{3}=2(3)^{2}-3H _{3}=15 | H_{4}=2(4)^{2}-4H _{4}=28 |

The table below shows each sequence’s formulas and the first five numbers.

Polygon | Formula | n=1 | n=2 | n=3 | n=4 | n=5 |

Heptagonal | P_{n} = $\frac{5n^2-3n}{2}$ | 1 | 7 | 18 | 34 | 55 |

Octagonal | P_{n} = 3n^{2 }– 2n | 1 | 8 | 21 | 40 | 65 |

Nonagonal | P_{n} = $\frac{7n^2-5n}{2}$ | 1 | 9 | 24 | 46 | 75 |

Decagonal | P_{n} = (2n)^{2} – 3n | 1 | 10 | 27 | 52 | 85 |

**Sequence on Polygonal Numbers**

**Triangular Number Sequence**

The triangular numbers follow these sequence:

1, | 3, | 6, | 10, | 15, | 21, | 28, | 36, | 45, | 55, | 66, | 78, | 91, |

105, | 120, | 136, | 153, | 171, | 190, | … |

a) The first triangular number is 1. b) The second triangular number: 1 + 2 = 3 c) The third triangular number: 1 + 2 + 3 = 6 d) The fourth triangular number is 1 + 2 + 3 + 4 = 10 e) So, any triangular number is equal to the sum of all positive integers up to and including its term number, or, in other words, Tn = 1 + 2 + 3 + 4 + … + n. Thus, T _{1}=1T _{2}=1+2=3T _{3}=1+2+3=6T _{4}=1+2+3+4=10f) A triangular number can be computed by the formula T _{n}=$\frac{n ( n+1 )}{2}$. |

**Square Number Sequence**

The square numbers follow these sequence:

1, | 4, | 9, | 16, | 25, | 36, | 49, | 64, | 81, | 100, | 121, | 144, | 169, |

196, | 225, | 256, | 289, | 324, | … |

a) The first square number is 1. b) The second square number is 1 + 3 = 4 c) The third square number is 1 + 3 + 5 = 9 d) The fourth triangular number is 1 + 3 + 5 + 7 = 16 e) Any square number is equal to the sum of n positive odd numbers, hence, Sn = 1 + 3 + 5 + 7 + … + n. f) Notice also that if we square the number position, for example, 1 ^{2}, 2^{2}, 3^{3}, 4^{2}We will get the square number sequence S _{1}=1^{2}=1S _{2}=2^{2}=4S _{3}=3^{2}=9S _{4}=4^{2}=16So, the next number for the fifth position is 5 ^{2 }=25.A square number can be computed by the formula S _{n}=n^{2}. |

**Pentagonal Number Sequence**

The pentagonal numbers follow these sequence:

1, | 5, | 12, | 22, | 35, | 51, | 70, | 92, | 117, | 145, | 176, | 210, | 247, |

287, | 330, | 376, | 425, | 477, | … |

a) The first pentagonal number is 1. b) The second pentagonal number is 1 + 4 = 5 c) The third pentagonal number is 1 + 4 + 7 = 12 d) The fourth pentagonal number is 1 + 4 + 7 + 12 = 16 e) A pentagonal number can be computed by the formula P _{n}= $\frac{( 3n^2-n )}{2}$ |

**Hexagonal Numbers**

The hexagonal numbers follow these sequence:

1, | 6, | 15, | 28, | 45, | 66, | 91, | 120, | 153, | 190, | 231, | 276, | 325, |

378, | 435, | 496, | 561, | 630, | … |

a) The first hexagonal number is 1. b) The second hexagonal number is 1 + 5 = 6 c) The third hexagonal number is 1 + 5 + 9 = 15 d) The fourth hexagonal number is 1 + 5 + 9 + 13 = 28 e) A hexagonal number can be computed by the formula H _{n}=2n^{2}-n |

**Examples**

**Example 1**

Find each of the following triangular numbers.

( a ) 20^{th} triangular number

( b ) 52^{nd} triangular number

( c ) 44^{th} triangular number

( d ) 100^{th} triangular number

*Solution:*

To get the missing triangular number, we will use this formula: T_{n}=$\frac{n ( n+1 )}{2}$

( a ) Since we are solving for the 20^{th} triangular number, we must use n = 20. So, we have,

T_{n}=$\frac{n ( n+1 )}{2}$

T_{20}=$\frac{20 ( 20+1 )}{2}$

T_{20}=$\frac{20 ( 21 )}{2}$

T_{20}=$\frac{420}{2}$

T_{20} = 210

Thus, the 20^{th} triangular number is 210.

( b ) We will use n =52 since the missing number is the 52^{nd} triangular number.

T_{n}=$\frac{n ( n+1 )}{2}$

T_{52}=$\frac{52 ( 52+1 )}{2}$

T_{52}=$\frac{52 ( 53 )}{2}$

T_{52}=$\frac{2756}{2}$

T_{52} = 1378

Thus, the 52^{nd} triangular number is 1378.

( c ) The number’s position, 44, must be substituted to n to get the 44^{th }triangular number.

T_{n}=$\frac{n ( n+1 )}{2}$

T_{44}=$\frac{44 ( 44+1 )}{2}$

T_{44}=$\frac{44 ( 45 )}{2}$

T_{44}=$\frac{1980}{2}$

T_{44}=990

Thus, the 44^{th} triangular number is 990.

( d ) Since we are solving for the 100^{th} triangular number, we must use n = 100. So, we have,

T_{n}=$\frac{n ( n+1 )}{2}$

T_{100}=$\frac{100 ( 100+1 )}{2}$

T_{100}=$\frac{100 ( 101 )}{2}$

T_{100} = $\frac{10100}{2}$

T_{100} = 5050

Thus, the 100^{th} triangular number is 5050.

**Example 2**

Find each of the following square numbers.

( a ) 25^{th} square number

( b ) 42^{nd} square number

( c ) 50^{th} square number

( d ) 65^{th} square number

*Solution:*

To find the square numbers, we will use the formula S_{n}=n^{2}

( a ) To find the 25^{th }square number, we must use n = 25. That is,

S_{n}=n^{2}

S_{25}=25^{2}

S_{25}=( 25 )( 25 )

S_{25} = 625

Thus, the 25^{th} square number is 625.

( b ) We will use n = 42 to find the 42^{nd} square number.

S_{n}=n^{2}

S_{42}=42^{2}

S_{42}= ( 42 )( 42 )

S_{42}=1764

Hence, the 42^{nd} square number is 1764.

( c ) To find the 50^{th }square number, we must use n = 50. That is,

S_{n}=n^{2}

S_{50}=50^{2}

S_{50}=( 50 )( 50 )

S_{50} = 2500

Thus, the 50^{th} square number is 2500.

( d ) We will use n = 65 to find the 65^{th} square number.

S_{n}=n^{2}

S_{65}=65^{2}

S_{65}= ( 65 )( 65 )

S_{65}=4225

Hence, the 65^{th} square number is 4225.

**Example 3**

Answer each.

( a ) Find the 8^{th} pentagonal number.

( b ) Which of the following is the 12^{th} pentagonal number? 200, 210, or 220?

( c ) Solve for the 25^{th} pentagonal number.

*Solution:*

Here is the formula to use in finding the n^{th }pentagonal number:* *P_{n} = $\frac{( 3n^2-n )}{2}$

( a ) To find the 8^{th} pentagonal number, we must use n = 8.

P_{n} = $\frac{( 3n^2-n )}{2}$

P_{8} = $\frac{( 3(8)^2-8 )}{2}$

P_{8}= $\frac{( 3(64)-8 )}{2}$

P_{8}= $\frac{( 192-8 )}{2}$

P_{8}= $\frac{184}{2}$

P_{8}=92

Therefore, the 8^{th} pentagonal number is 92.

( b ) To find the 12^{th} pentagonal number, we must use n = 12.

P_{n} = $\frac{( 3n^2-n )}{2}$

P_{12} = $\frac{( 3(12)^2-12 )}{2}$

P_{12}= $\frac{[ 3 ( 144 )-12 ) ]}{2}$

P_{12}= $\frac{( 432-8 )}{2}$

P_{12}= $\frac{420}{2}$

P_{12}=210

Hence, the 12^{th} pentagonal number is 210.

( c ) The value of n must be 25 to find the 25^{th} pentagonal number.

P_{n} = $\frac{( 3n^2-n )}{2}$

P_{25} = $\frac{( 3(25)^2-25 )}{2}$

P_{25}= $\frac{[ 3 ( 625 )-25 ) ]}{2}$

P_{25}= $\frac{( 1875-25 )}{2}$

P_{25}= $\frac{1850}{2}$

P_{25}=925

Thus, the 25^{th} pentagonal number is 925.

**Example 4**

Find the missing hexagonal numbers.

1, | 6, | 15, | 28, | 45, | 66, | 91, | 120, | ______ | 190, | 231, | 276, | 325, |

378, | ______ | 496, | 561, | 630, | ______ | 780, | 861, | 946, | 1035, | 1128, | 1225, | ______ |

1431, | 1540, | 1653, | 1770, | 1891, | ______ | … |

*Solution:*

From the given sequence of numbers, we are looking for the 9^{th},15^{th}, 19^{th}, 26^{th}, and 32^{nd} hexagonal numbers.

Let us use the formula H_{n}=2n^{2}-n to get the value of the missing hexagonal numbers.

For the 9^{th} hexagonal number, we have,

H_{n}=2n^{2}-n

H_{9}=2( 9 )^{2} – 9

H_{9}=2( 81 ) – 9

H_{9}=162-9

H_{9} = 153

Thus, the 9^{th} hexagonal number is 153.

For the 15^{th} hexagonal number,

H_{n}=2n^{2}-n

H_{15}=2( 15 )^{2} – 15

H_{15}=2( 225 )-15

H_{15} = 450-15

H_{15} = 435

Thus, the 15^{th} hexagonal number is 435.

To find the 19^{th} hexagonal number, we must use n = 19.

H_{n}=2n^{2}-n

H_{19}=2( 19 )^{2} – 19

H_{19}=2( 361 )-19

H_{19}=722-19

H_{19}=703

Therefore, the 19^{th} hexagonal number is 703.

To find the 26^{th} hexagonal number, we must use n = 26.

H_{n}=2n^{2}-n

H_{26}=2( 26 )^{2} – 26

H_{26}=2( 676 )-26

H_{26}=1352-26

H_{26}=1326

Therefore, the 26^{th} hexagonal number is 1326.

For the 32^{nd} hexagonal number, we have,

H_{n}=2n^{2}-n

H_{32}=2( 32 )^{2} – 32

H_{32}=2( 1024 )-32

H_{32}=2048-32

H_{32}=2016

Thus, the 32^{nd} hexagonal number is 2016.

Here is the list with complete answers for the missing numbers.

1, | 6, | 15, | 28, | 45, | 66, | 91, | 120, | 153, | 190, | 231, | 276, | 325, |

378, | 435, | 496, | 561, | 630, | 703, | 780, | 861, | 946, | 1035, | 1128, | 1225, | 1326, |

1431, | 1540, | 1653, | 1770, | 1891, | 2016, | 2145, | 2278, | 2415, | 2556, | 2701, | 2850, | 3003, |

**Example 5**

Find the value of the sum of the 12^{th} triangular and the 13^{th} pentagonal numbers.

*Solution:*

Let us first get the values of the 12^{th} triangular and the 13^{th} pentagonal numbers.

For the 12^{th} triangular number, n = 12.

T_{n}=$\frac{n ( n+1 )}{2}$

T_{12}=$\frac{12 ( 12+1 )}{2}$

T_{12}=$\frac{12 ( 13 )}{2}$

T_{12}=$\frac{156}{2}$

T_{12}=78

Thus, the 12^{th} triangular number is 78.

For the 13^{th} pentagonal number, the value of n must be 25.

P_{n}= $\frac{( 3n^2-n )}{2}$

P_{13}= $\frac{[ 3(13)^2-13 ]}{2}$

P_{13}= $\frac{[ 3( 169 )-13 ) ]}{2}$

P_{13}= $\frac{( 507-13 )}{2}$

P_{13}= $\frac{494}{2}$

P_{13}=247

Hence, the 13^{th} pentagonal number is 247.

To answer the question, let us now add 78 and 247 as the 12^{th} triangular and the 13^{th} pentagonal numbers, respectively. So, we have,

78+247=325

Therefore, the sum of the 12^{th} triangular number and the 13^{th} pentagonal number is 325.

**Example 6**

Find the missing numbers in each polygonal number sequence.

( a ) 1, 3, 6, 10, 15, 21, **_**_, 36, 45, 55, 66, 78, 91, __, 120, 136, 153,

(b ) 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, __, 144, 169, 196, __, 256,

( c ) 1, 5, 12, 22, 35, 51, __, 92, 117, 145, 176, 210, 247, 287, __, 376,

( d ) 1, 6, 15, 28, 45, 66, 91, 120, 153, 190, __** , **276, 325, 378, 435, __, 561

*Solution:*

( a ) The number pattern shows triangular numbers. The 7^{th} and 14^{th} triangular numbers are missing. To get the values of each, let us use this formula: T_{n}=$\frac{n ( n+1 )}{2}$

For the 7^{th} triangular number, we must use n = 7 while n = 14 for the 14^{th} triangular number.

T_{n}=$\frac{n ( n+1 )}{2}$T _{7}=$\frac{7 ( 7+1 )}{2}$T _{7}=$\frac{7( 8 )}{2}$T _{7}=$\frac{56}{2}$T _{7}=28 | T_{n}=$\frac{n ( n+1 )}{2}$T _{14}=$\frac{14 ( 14+1 )}{2}$T _{14}=$\frac{14( 15 )}{2}$T _{14}=$\frac{210}{2}$T _{14}=105. |

Thus, the 7^{th} and 14^{th} triangular numbers are 28 and 105, respectively.

( b ) The set shows the square numbers. We will use the formula: S_{n}=n^{2} to find the 11^{th} and 15^{th} square numbers.

S_{n}=n^{2}S _{11}=11^{2}S _{11}=( 11 )( 11 )S _{11} = 121 | S_{n}=n^{2}S _{15}=15^{2}S _{15}=( 15 )( 15 )S _{15} = 225 |

Hence, the 11^{th} square number is 121 while the 15^{th} is 225.

( c ) The sequence’s 7^{th} and 13^{th} pentagonal numbers are missing. Let us use the formula: P_{n}= $\frac{( 3n^2-n )}{2}$

P_{n}= $\frac{( 3n^2-n )}{2}$P _{7}= $\frac{[ 3(7)^2-7 ]}{2}$P _{7}= $\frac{[ 3( 49 )-7 ) ]}{2}$P _{7}= $\frac{( 147-7 )}{2}$P _{7}=$\frac{140}{2}$P _{7}=70 | Pn= $\frac{( 3n^2-n )}{2}$ P _{15}= $\frac{[ 3(15)^2-15 ]}{2}$P _{15}= $\frac{[ 3( 225 )-15 ) ]}{2}$P _{15}= $\frac{( 675-15 )}{2}$P _{15}= $\frac{660}{2}$P _{15}=330 |

Hence, the 7^{th} square number is 70 while the 15^{th} is 330.

( d ) The number pattern shows hexagonal numbers. To get the 11^{th} and 16^{th} hexagonal numbers, let us use this formula: H_{n}=2n^{2}-n

H_{n}=2n^{2}-nH _{11}=2( 11 )^{2}-11H _{11}=2( 121 )-11H _{11}=242-11H _{11}=231 | H_{n}=2n^{2}-nH _{16}=2( 16 )^{2}-16H _{16}=2( 256 )-16H _{16}=512-16H _{16}=496 |

Therefore, the 11^{th} and 16^{th} hexagonal numbers are 231 and 496.

**Summary**

Polygonal numbers are nonnegative integers that are represented by regular polygons made of geometrically arranged, evenly spaced points. When the number of equidistant points used to represent a polygon expands, it enlarges in size in a predictable way.

We can find the next polygonal number using dots. The augmenting array of dots of polygonal numbers has the following stages.

The first polygonal number is always 1.

The polygon’s vertex count is the second number.

The third and subsequent values can be obtained by adding one point to each of two adjacent sides, followed by creating another regular polygon on top of the first.

The most common polygonal numbers are triangular numbers, square numbers, pentagonal numbers, and hexagonal numbers.

The first 15 triangular numbers are:

1, | 3, | 6, | 10, | 15, | 21, | 28, | 36, | 45, | 55, | 66, | 78, | 91, | 105, | 120 |

The first 15 square numbers are:

1, | 4, | 9, | 16, | 25, | 36, | 49, | 64, | 81, | 100, | 121, | 144, | 169, | 196, | 225 |

The first 15 pentagonal numbers are:

1, | 5, | 12, | 22, | 35, | 51, | 70, | 92, | 117, | 145, | 176, | 210, | 247, | 287, | 330 |

The first 15 hexagonal numbers are:

1, | 6, | 15, | 28, | 45, | 66, | 91, | 120, | 153, | 190, | 231, | 276, | 325, | 387, | 435 |

Formulas in getting the n^{th } polygonal numbers, where n is any natural number,

Triangular Numbers: T_{n}=$\frac{n ( n+1 )}{2}$

Square Numbers:* *S_{n}=n^{2}

Pentagonal Numbers: P_{n}= $\frac{( 3n^2-n )}{2}$

Hexagonal Numbers: H_{n}=2n^{2}-n

**Frequently Asked Questions on Polygonal Numbers (FAQs)**

**What is meant by polygonal numbers?**

Polygonal numbers are nonnegative integers that are represented by regular polygons made of geometrically arranged, evenly spaced points. When the number of equidistant points used to represent a polygon expands, it enlarges in size in a predictable way.

**What is the next polygonal number?**

In finding the n^{th} term of polygonal numbers, certain formulas depend on the sequence you are dealing with. The most common set of polygonal numbers is the triangular and square numbers.

Formulas in getting the n^{th } polygonal numbers, where n is any natural number,

Triangular Numbers: T_{n}=$\frac{n ( n+1 )}{2}$

Square Numbers:* *S_{n}=n^{2}

Pentagonal Numbers: P_{n}= $\frac{( 3n^2-n )}{2}$

Hexagonal Numbers: H_{n}=2n^{2}-n

**How do we use dots to find the next polygonal number?**

We can find the next polygonal number using dots. The augmenting array of dots of polygonal numbers has the following stages.

The first polygonal number is always 1.

The polygon’s vertex count is the second number.

The third and subsequent values can be obtained by adding one point to each of two adjacent sides, followed by creating another regular polygon on top of the first.

The first three triangular numbers.

The first three square numbers.

The first three pentagonal numbers.

The first three hexagonal numbers.

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