Geometry: An Interactive Journey to Mastery

Lesson 34: The Geometry of Figurate Numbers

Example 1

Albert buys one piece of candy on January 1, two pieces on January 2, etc., all the way up to 365 pieces on

December 31. How many total pieces of candy did Albert buy that year?

Solution

Albert bought 1 2 3  " 365 365 366u 2 66, 795 pieces of candy.

Example 2

If Tͼ௘N௘ͽGHQRWHVWKHN th triangle number, show that Tͼ௘N + M௘ͽ Tͼ௘N௘ͽTͼ௘M௘ͽN × M.

Solution

Figure 34.1UHSUHVHQWVVWDFNLQJWZRWULDQJOHVͼ௘RQHZLWKD

base row of N dots and one with a base row of MGRWV௘ͽDQG

a rectangle to create a triangular array of dots with a base row

of N + M dots.

Figure 34.2 is the diagram for N = 6 and M = 4 with the dots

drawn in.

Algebraically, we have the following.

22

22

1

2 2

22

22 2 11

(^22).

TN M NMNM

N MNM NM

NN MNMM

NN MN MM

TN MN TM

 









ͼ௘7KLVDOJHEUDLFDSSURDFKLVOHVVHQOLJKWHQLQJ௘ͽ

M

M

N

N

Figure 34.1

4

6

6 4

Figure 34.2