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New Topics for Junior High (2006) 1

Figurate Numbers

Figurate numbers is a general name given to any class or

of numbers that are formed and represented by geometric figures. You •••• •••• •••• are already familiar is one class of figurate numbers: the squares. Square •••• numbers get their name from their geometric representation as a square. 42 = 16 For example, the fourth is 16 and can be drawn with a 4 × 4 square consisting of 16 dots.

Triangular Numbers The first class of figurate numbers, however, is the triangular numbers. As their name implies, the triangular numbers are the numbers that can be represented by . Many people use equilateral triangles to show the triangular numbers. In this paper, we use isosceles right triangles. The first few triangular numbers are 1, 3, 6, 10, 15, 21, . . . .

• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • ... 1 3 6 10 15

To form the nth , you take a column with 1 dot, a column with 2 dots, a column with 3 dots, etc., up to a column with n dots. This forms the isosceles right shape. Therefore, the number of dots in the figure is equal to the nth triangular number. If we let Tn denote the nth triangular number, then we have Tn = 1 + 2 + 3 + ... + n.

There is a nice short formula for Tn. To derive it, observe the following example. Notice that in the figure, we take two copies of • ...... the triangles representing T4, and arrange them so that they form •• ...... a rectangle whose are 4 × 5. The area of this rectangle ••• ...... •••• ..... is equal to the total number of dots and is 20. Since this represents ...... twice T4, we have 2T4 = 20 which gives T4 = 10.

You should convince yourself that when you perform this construction in general for Tn, n(n + 1) the rectangle formed has dimensions n × (n + 1) and thus T = . n 2

1This document contains information pertaining to a new topic that will be covered on PSIA Mathematics tests starting in 2006. The goal of the new topics page is to provide basic information to motivate students to research and discover ideas related to this topic. PSIA – 2006 – New Topics for Junior High Mathematics — Page 2

Square Numbers • ...... The square numbers, denoted by Sn, are formed by squares. How- •• ...... ever, the square representing Sn can be decomposed into two tri- ••• ......

angular numbers: Tn and Tn−1. Therefore, we have the equation •••• 2 Sn = Tn−1+Tn. The figure on the right show that 4 = S4 = T3+T4.

Other Two-dimensional Figurate Numbers Given any n, n ≥ 3, figurate numbers based on n ...... •...... exist. These are in general formed by creating regular poly- ...... •... ..•...... gons, sharing the same initial dot, with sides from 1 to n. An ...... •... ..•...... •...... •...... •...... example of P4, the fourth , is shown...... •. •...... Although it is not immediately obvious, all two- •...... •. ..•...... •...... dimensional figurate numbers, also called polygonal numbers, .•...... •... ..•. ..•...... can be broken down into the sum of triangular numbers. .•...... •...... •...... •.. Moreover, the general formula for the nth polygonal num-

ber with r sides is Tn + (r − 3) · Tn−1. In other words, it is the sum of the nth triangular number and r − 3 copies of the (n − 1)th triangular number.

Certainly, for some polygonal numbers, simpler formulas exist. For example, letting Hexn

denote the nth , we have the following: Hexn = Tn + (6 − 3)Tn−1 = n(n + 1) 3n(n − 1) n2 + n + 3n2 − 3n 4n2 − 2n + = = = 2n2 − n = n(2n − 1). 2 2 2 2 Tetrahedral Numbers The tetrahedral numbers are represented geometrically as . ..•...... Each layer of the is a triangle and in fact, the mth layer of .•...... •...... •...... the tetrahedron has Tm dots. The first few tetrahedral numbers are 1, 4, ...... •...... •.. •...... •.... 10, 20, 35, 56, . . . . •...... •......

Exercises2

In the following exercises, let Tn denote the nth triangular number, Sn the nth square number, Pn

the nth pentagonal number, and Hexn the nth hexagonal number.

1. Find the following numbers: T7, T12, T1000, S9, S100, P3, P6, Hex4, Hex50.

2. Write S12 as the sum of two triangular numbers.

3. Aside from the number 1, can Sn = Tm for some two different numbers n and m? If so, when?

4. If n is a fixed integer greater than 1, which is larger: Pn or Sn?

5. Find T etra5, where T etran is the nth . Sketch the appropriate figure.

6. Define square (denoted SPn) similarly to the definition of tetrahedral

number. How many dots are on the mth layer of of SPn for 1 ≤ m ≤ n?

2These exercises are merely provided to give the student and teacher a beginning point for discussion of the topics listed in the paper.