patterns, number of trapezoid in a layered triangle total number of trapezoid in a triangle formula

1. Republic of the Philippines
SORSOGON STATE COLLEGE
Teacher – Education Department
Sorsogon City Main Campus
2nd Semester, AY 2016 – 2017
NAME:NONILYN D. DIAZ
COURSE/SECTION: BSED MATH 3A
DESCRIPTION OF THE STUDY
Patterns are all over us. It helps us to predict of what will be the next in the sequence. In
this study, pattern was used to find the total number of trapezoids in an Nlayered – triangle. It
includes all the trapezoid formed in a triangle (Inverted, small – big and Upright, small – big). In
order to look for a pattern, different ways of counting has been conducted.
PRESENTATION OF THE STUDY
How do we make an Nlayered – triangle?
In making an Nlayered – triangle, we simply add triangles from the base of the given
triangle to complete the desired layer. For example, we want to make a 2layered – triangle. If we
have 1layered – triangle as seen in figure 1, then let’s add another triangles to complete the next
layer. That’s one way of making an Nlayered – triangle.
Since we are already know how to make an Nlayered – triangle, we are going to count the
number of trapezoids formed in that Nlayered – triangle.
Figure 1
Figure 2

2. Figure 2. 2Layered Triangle
ngle
Figure 1. 1Layered Triangle
In Figure 1,
obviously we don’t
have trapezoid
formed.
In Figure 2, we have
3 trapezoids formed. 3 × 1
In Figure 2, we have
18 trapezoids
formed.
In Figure 2, we have
57 trapezoids
formed.
3 × 6
3 × 19

3. As we can see, in figure 4 we already have 57 trapezoids formed. We are now expecting a large
number of trapezoids for the next figure. So, what we are going to do is that we are going to
focus only on one part of the triangle and we are going to separate the counting of upright
from inverted trapezoids. We are also going to count the trapezoid layer by layer.
Consider this Tra – tri figure, the number from the right side represents the total number of
upright trapezoids by layer and from the left side represents the Nlayered triangle.
e
0
3
1
6
10
15
21
28
1
2
3
4
5
6
7
8

4. Based from the figure above, we can already see a pattern. In that way it can easily for us to
predict what’s next to 28 and that is 36.
Using the Tra – tri figure, we can easily count the upright number of trapezoid according to their
Nlayered.
Table 1
N U1 U2 U3 U4 U5 U6 U7 U8
1 0
2 1 0
3 1+3 1 0
4 1+3+6 1+3 1 0
5 1+3+6+10 1+3+6 1+3 1 0
6 1+3+6+10+15 1+3+6+10+15 1+3+6 1+3 1 0
7 1+3+6+10+15+21 1+3+6+10+15+21 1+3+6+10+15 1+3+6 1+3 1 0
8 1+3+6+10+15+21+28 1+3+6+10+15+21+28 1+3+6+10+15+21 1+3+6+10+15 1+3+6 1+3 1 0
Table 1 shows how to use the Tra – tri Figure in counting upright trapezoid.
U1 = One – layered upright trapezoidU2 = Two – layered upright trapezoid
U3 = Three – layered upright trapezoid and so on.
Table 2 (Total of Upright Trapezoid)
TOTAL N U1 U2 U3 U4 U5 U6 U7 U8
0 1 0
1 2 1 0
5 3 4 1 0
15 4 10 4 1 0
35 5 20 10 4 1 0
70 6 35 20 10 4 1 0
126 7 56 35 20 10 4 1 0
210 8 84 56 35 20 10 4 1 0
And for inverted trapezoid, we are also going to use the Tra – tri figure for counting.
0
3
1
6
10
1
2
3
4

5. As you can see in 2Layered triangle, there is no inverted triangle. Inverted triangles can be seen
from 3Layered triangle to NLayered triangle. How do we count inverted trapezoids? Using
3Layered triangle, as you can see the inverted trapezoid in the red mark; try to look its reverse
and its corresponding number. In 3Layered triangle, we have 1 inverted trapezoid. Using the
same process, we have the following data below:
Table 3 (Inverted Trapezoid)
N I1 I2 I3 I4
1 0
2 0
3 1 0
4 1+3 0
5 1+3+6 1 0
6 1+3+6+10 1+3 0
7 1+3+6+10+15 1+3+6 1 0
8 1+3+6+10+15+21 1+3+6+10 1+3 0
Table 3 (Inverted Trapezoid- TOTAL)
N I1 I2 I3 I4 TOTAL
1 0 0
2 0 0
3 1 0 1
4 4 0 4
5 10 1 0 11
6 20 4 0 24
7 35 10 1 0 46
8 56 20 4 0 80
With the use of Tra – tri Figure, we can already get the total number of trapezoid formed in
Nlayered Triangle.
TNT = 3 (Upright + Inverted)
TNT = 3 (U +I)

6. Given this data we are going to make formula for finding the total number of trapezoid formed
in Nlayered Triangle.
FINITE DIFFERENCE FOR UPRIGHT TRAPEZOID
N 1 2 3 4 5 6
TOTAL 0 1 5 15 35 70
𝑎𝑛4 + 𝑏𝑛3 + 𝑐𝑛2 + 𝑑𝑛 + 𝑒
𝑎 =
1
24
𝑏 =
1
12
𝑐 = −
1
24
𝑑 = −
1
12
𝑈 =
1
24
𝑛4 +
1
12
𝑛3 −
1
24
𝑛2 −
1
12
𝑛
𝑈 =
𝑛4+2𝑛3−𝑛2−2𝑛
24
FINITE DIFFERENCE FOR INVERTED TRAPEZOID
N 3 4 5 6 7 8
TOTAL 1 4 11 24 46 80
𝑎𝑛4 + 𝑏𝑛3 + 𝑐𝑛2 + 𝑑𝑛 + 𝑒
1 4 10 20 35
3 6 10 15
3 4 1
511
3 7 13 22 34
4 6 9 12
432
1 1

7. 𝑎 =
1
24
𝑏 = −
5
12
𝑐 =
71
24
𝑑 = −
115
12
𝑒 = 11
𝐼 =
1
24
𝑛4 −
5
12
𝑛3 +
71
24
𝑛2 −
115
12
𝑛 + 11
𝐼 =
𝑛4−10𝑛3+71𝑛2−230𝑛
24
+ 11
Substituting the values for TNT = 3 (U +I), we get
TNT = 𝟑(
𝑛4+2𝑛3−𝑛2−2𝑛
24
+
𝑛4−10𝑛3+71𝑛2−230𝑛
24
+ 11)
TNT = 𝟑(
𝑛4+2𝑛3−𝑛2−2𝑛+𝑛4−10𝑛3+71𝑛2−230𝑛
24
+ 11)
TNT = 𝟑(
2𝑛4−8𝑛3+70𝑛2−232𝑛
24
+ 11)
TNT =
2𝑛4−8𝑛3+70𝑛2−232𝑛
8
+ 33
TNT = 2 (
𝑛4−4𝑛3+35𝑛2−116𝑛
8
) + 33
TNT =
𝑛4−4𝑛3+35𝑛2−116𝑛
4
+ 33
TNT =
𝒏( 𝑛3−4𝑛2+35𝑛−116)
𝟒
+ 33
TNT = 𝑛 (
𝑛3+35𝑛
𝟒
− 𝑛2 − 29) + 33
FINAL FORMULA
TNT = 𝑛 (
𝑛3+35𝑛
𝟒
− 𝑛2 − 29) + 33
Where n = number of layers in triangle (NLayered Triangle)
TNT = Total number of trapezoid in an NLayered Triangle