A pentagon number is a figurate number that represents a pentagon. A pentagon has five sides, so it follows that all pentagonal numbers have the value of 5. The first few pentagonal numbers are given by 1, 5, 12, 22, 35, 51, 70, 92 and 118. There are several different ways of writing pentagonal numbers.

1 5 12 22 35 51 70 92 117 145

1. The first ten pentagonal numbers are: 1, 5, 12, 22, 35, 51, 70, 92, 117 and 145.
2. A pentagonal number is the sum of the first n triangular numbers.
3. The sum of the first n pentagonal numbers is equal to (n2 + 11n).

1,5

• The first ten pentagonal numbers are 1, 5, 12, 22, 35, 51, 70, 92, 117 and 143.
• The first few terms of the sequence are given by the formula \(10^{\lceil n/2\rceil}\) where \(n\) is an integer greater than or equal to 2. For example:
• (10^{\lceil 3/2\rceil} = 5; 10^{\lceil 4/2\rceil} = 12; 10^{\lceil 5/2\rceil} = 22\) and so on.

The name comes from the fact that they are the number of sides a regular polygon would have if it had n vertices. Hence they are also known as pentagonal numbers (since a regular pentagon has five sides).

12,55

The first ten pentagonal numbers are:

• 1, 5, 12, 55, 156, 325, 462, 635 and 756.
• The pentagonal number 5 is the only one that ends in 5. The next one is 12 (1+2=3+4=5). Then we have 55 (5+6=7+8=9+10=11+12=13), 156 (13+14=15+16=17+18=19+20=21) and so on until 756 (26+27=28+29=30).

22,65

The first ten pentagonal numbers are:

• 1, 5, 12, 22, 35, 51, 70, 92, 117, 145

The next ten are:

• 167, 205, 240, 271, 298, 323, 346, 365, 384

35,70

The first ten pentagonal numbers are:

• 1, 5, 12, 22, 35, 50, 66, 78, 91
• The pattern of these numbers is as follows: 1, 3 + 2 = 5; 2 + 3 + 4 = 12; 3 + 4 + 5 = 22; and so on. In general: n(n+1)/3 = (n+1)2/3-1

51,75

• The first ten pentagonal numbers are: 1, 5, 12, 22, 35, 51, 75, 101, 126 and 151.
• In mathematics and number theory, a pentagonal number is a figurate number that can be represented by a regular pentagon. The first few pentagonal numbers are 1 (the smallest), 5, 12, 22

70,80

The first ten pentagonal numbers are:

• 1, 5, 12, 22, 35, 51, 70, 80.
• A pentagonal number is a figurate number that represents the number of dots in a regular pentagon. The first few pentagonal numbers are 1, 5, 12 and 22.

The sum of the first n pentagonal numbers is given by:

• The sum of the first n pentagonal numbers can be expressed as:
• The first ten pentagonal numbers are: 1, 5, 12, 22, 35, 51 70 and 80.

92,85

• 1, 5, 12, 22, 35, 49, 65, 78, 91

The first ten pentagonal numbers are:

• 1, 5, 12, 22, 35, 49, 65 and 91.
• A pentagonal number can be written as the sum of consecutive integers from 1 to n (the nth term of an arithmetic sequence). The first few examples are 1 + 3 + 6 + 10 = 20; and 2 + 4 + 6 + 10 = 20. The first few pentagonal numbers are 1, 5, 12 and 20.

The formula for the nth term of a pentagonal number is:

• n(n+2)/2

117,90

The first ten pentagonal numbers are:

• 1, 5, 12, 22, 35, 49, 66, 85, 102
• A pentagonal number is a figurate number that represents the number of dots in a five-sided polygon. The first pentagonal number is 1 and each succeeding pentagonal number is one more than the previous one.

The first ten pentagonal numbers are:

• 1, 5, 12, 22, 35, 49, 66, 85 and 102

The first ten pentagonal numbers are 1,5; 12,55; 22,65; 35,70; 51,75; 70,80; 92,85; 117,90.

• The first few numbers of the sequence are 1,5; 5,12; 12,22; 22,35; 35,51; 51,70; 70,92; 92,117
• The next term is 122 = 132 – 2 = 172 – 2 = (1 + 2)2 – 2 = 5 – 2
• The pentagonal numbers are the numbers that result from adding the first n natural numbers together. For example, 1+2+3+4+5=15. The first ten pentagonal numbers are 1,5; 12,55; 22,65; 35,70; 51,75; 70,80; 92,85; 117,90.
• The sum of these ten numbers is 379.
• The triangle of first n pentagonal numbers is a sequence of triangular arrays that sums to n²:
• 1 + 5 = 6
• 12 + 55 = 67
• 22 + 65 = 87
• 35 + 70 = 105
• 51 + 75 = 126 (1)

Conclusion

Pentagonal numbers are base 10 numbers that are pentagonal. Their form is the shape of a pentagon with sides of equal length. The first few are 1, 5, 12, 22, 35, 51 and 70. The pentagonal numbers usually appear in areas of mathematics. One formula for these numbers is: {10[(n-4)(n+1)(2n -1) ÷ 2]}. In general all numbers can be represented by (a^2 +2b^2 +c^2). A number like 674=400+349+84, therefore it’s equivalent to the equation {10[{10[3(1-4)]+5}{10[5(5)+5]}÷2]}. Since a, b and c will always be non negative integers hence the equation can never be 0.