This document discusses different types of figurate numbers including triangular, square, pentagonal, and hexagonal numbers. It provides examples of the patterns formed by each type of number and formulas to calculate the nth term. Relations between different figurate numbers are also explored, such as how the 4th square number can be expressed as the sum of the 3rd and 4th triangular numbers. A brief history of polygonal numbers is given, noting they were first studied by Pythagoras.

1. Chapter 3 FigurateNumbers Prepared by: Nathaniel T. Sullano BS Math - 3

2. A figurate number, also known as a figural number or polygonal number, is a number that can be represented by a regular geometrical arrangement of equally spaced points. source: http://mathworld.wolfram.com/FigurateNumber.html

3. Example: Triangular Square Pentagonal Hexagonal Image source: http://mathworld.wolfram.com/images/eps-gif/PolygonalNumber_1000.gif

4. Square Numbers represented by squares 1st 2nd 3rd 4th 5th Image source: http://www.learner.org/courses/mathilluminated/images/units/1/1094.gif

5. We can also view the square numbers from a different aspect. Like the figure below Image source: http://tonks.disted.camosun.bc.ca/courses/psyc290/figure1002.jpg We can observe that a square is partitioned into a smaller square and a carpenter’s square or gnomon.

6. In observing the figure on the left, we see that the fourth square number is the sum of the first four odd numbers, 1 + 3 + 5 + 7 = 42 n2= (n – 1)2+(2n – 1) Image source: http://tonks.disted.camosun.bc.ca/courses/psyc290/figure1002.jpg

7. Below shows the first few square numbers as sums of odd numbers: 1st 1 = 12 2nd 1 + 3 = 4 = 22 3rd 1 + 3 + 5 = 32 4th 1 + 3 + 5 + 7 = 42 5th 1 + 3 + 5 + 7 + 9 = 52 6th 1 + 3 + 5 + 7 + 9 + 11 = 62

8. Generalizing, we see that the nth square number is the sum of the first n odd numbers, 1 + 3 + 5 + 7 + … + (2n – 1) The difference between any two consecutive addends is 2. Since the difference between any two consecutive numbers in the sum which names a square number is 2, we say the common difference of the square numbers is 2.

9. Triangular Numbers being pictured as triangles Image source: http://mathforum.org/workshops/usi/pascal/images/triang.dots.gif

10. If we observe the pictures above of the representations of the triangular numbers, we see that the number of dots in the representation of the first triangular number is 1, of the second, 1 + 2, of the third, 1 + 2 + 3, and so on. Image source: http://mathforum.org/workshops/usi/pascal/images/triang.dots.gif

11. 1st 1 2nd 1 + 2 3rd 1 + 2 + 3 4th 1 + 2 + 3 + 4 5th 1 + 2 + 3 + 4 + 5 . . . nth 1 + 2 + 3 + . . . + n where n is any counting number. Notice that the difference between any two addends in this sum is 1, so, 1 is the common difference of the triangular numbers. nth triangular number is given by the formula,

12. Pentagonal Numbers pictured in the form of a pentagon represented by a square with a triangle on top Image source: http://thm-a02.yimg.com/nimage/17d300a4d2a62fa0

13. In general, the nth pentagonal number is

14. Refer to the figure below. We see that the fourth pentagonal number is made up of the fourth square number and the third triangular number. Solution:

15. 1st 1 = 1 2nd 1 + 4 = 5 3rd 1 + 4 + 7 = 12 4th 1 + 4 + 7 + 10 = 22 5th 1 + 4 + 7 + 10 + 13 = 35 6th 1 + 4 + 7 + 10 + 13 + 16 = 51 . . . nth 1 + 4 + 7 + … + (3n – 2) = Common difference of the addends is 3.

16. Following the triangular, square, and pentagonal numbers are the figurate numbers with differences: 4, 5, 6, and so on. If the common difference is 4, we have the hexagonal numbers. 1st 1 = 1 2nd 1 + 5 = 6 3rd 1 + 5 + 9 = 15 4th 1 + 5 + 9 + 13 = 28 . . . nth 1 + 5 + 9 + . . . + (4n – 3) = n(2n – 1)

17. Patterns from Figurate Numbers

18. Notice that the representation of the fourth square number is partitioned so that it is composed of the 3rd and 4th triangular number . Thus, 6 + 10 = 16.

19. Relation of Square and Triangular Numbers __________________________________________ Triangular number 1 3 6 10 ... Triangular number 1 3 6 … Square number 1 4 9 16 ... Table 3.2

20. Discoveries of Relations of Figurate and Ordinary numbers

21. Every whole number is the sum of three or less triangular numbers. For Example: 17 = 1 + 6 + 10 26 = 1 + 10 + 15 46 = 10 + 36 150 = 6 + 66 + 78 64 = 28 + 21 + 15 25 = 10 + 15 II. Every whole number is the sum of four or less square numbers. For Example: 56 = 36 + 16 + 4 = 62 + 42 + 22 150 = 100 + 49 + 1 = 102 + 72 + 12

22. III. If we multiply each triangular number by 6, add a plus the cube of which triangular number it is, say we get the kth, we get (k + 1)3 Example: 3rd triangular number, (6 x 6) + 1 + 33 = (3 + 1)3 In general,

23. Eight (8) times any triangular number add 1 is a square. Example, (8 x 1) + 1 = 9 = 32 (8 x 3) + 1 = 25 = 52 (8 x 6) + 1 = 49 = 72 In general,

24. Brief History of Polygonal Numbers

25. The theory of polygonal numbers goes back to Pythagoras, best known for his Pythagorean Theorem. Pythagoras 572 – 500 B.C. Image source: http://9waysmysteryschool.tripod.com/sitebuildercontent/sitebuilderpictures/pythagoras.jpg

26. End of Presentation Thanks for listening 