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Skip counting Number sense Number patterns Shape patterns SequencesHere you will learn about triangular numbers, including how to identify them and work with them in numerical and pictorial sequences. You will also learn how to find triangular numbers and determine whether a number is a triangular number using the nth term.
Students first learn about triangular numbers when they learn about sequences and series in high school math classes such as Algebra I and Algebra II.
Triangular numbers are numbers that can be represented by a pattern of dots arranged in triangle (equilateral triangle) with the same number of dots on each side of the triangle. The number of dots form a sequence known as triangular numbers.
The total number of dots in each triangular pattern is the triangular number. Therefore, 1, \, 3, \, 6, and 10 are the first four triangular numbers.
If this pattern were to continue, how many dots would the side of the next triangle have? What would be the total number of dots the triangle has?
The next triangle is below. There are 5 dots that make up each side of the triangle and there are a total of 15 dots that make up the triangle.
So, the next triangular number is 15. Notice each of the row of dots starting from the bottom and moving to the top (5 dots, 4 dots, 3 dots, 2 dots, 1 dot).
Do you notice a pattern from triangle to triangle or from total amount of dots to total amount of dots?
1 \hspace{0.9cm} 3 \hspace{0.9cm} 6 \hspace{0.9cm} 10 \hspace{0.9cm} 15 \; β¦From number to number in this sequence, you can draw the triangle with dots or look for another pattern. In this case, the pattern is +2, \, +3, \, +4, \, +5, etc.
Meaning, 1+2=3, \, 3+3=6, \, 6+4=10, \, 10+5=15
If you did not draw the next triangle in the sequence and used the pattern identified above, what do you think the total number of dots would be?
β21β would be the next triangular number because adding 6 to 15 is 21. If the triangle were to be drawn each side of the triangle would have 6 dots.
15+6=\bf{21}Continuing the pattern, you would get,
This produces the sequence of triangular numbers,
1, \, 3, \, 6, \, 10, \, 15, \, 21, \, 28, \, 36, \, 45, \, 55, \, 66, \, 78, \, 91, \, 105, \, 120, \, β¦
To determine the next triangular number in a numerical sequence, when given the sequence, we need to find the difference between the previous two terms and add one more than this value.
What if you were asked to find the 25 th triangular number in the sequence? Would you want to sketch out 25 βdotβ triangles? The answer is βno.β
Sketching out 25 triangles is not an efficient way to find out the 25 th triangular number. However, there is a formula you can use to calculate any triangular number.
This formula is also known as finding the βnthβ term of a sequence (or nth triangular number) where T_{n} is the number in the sequence you are looking for and βnβ is the term in the sequence.
So, in this case, finding the 25 th term, you would make βnβ equal to 25 to calculate it.
T_{n}=\cfrac{1}{2} \, n(n+1).
\begin{gathered}T_n=\cfrac{1}{2} \, (25)(25+1) \\\\ T_n=325 \end{gathered}
The 25 th term in the triangle number sequence is 325.
Use this quiz to check your grade 4 to 5 studentsβ understanding of number patterns. 10+ questions with answers covering a range of 4th and 5th grade number patterns topics to identify areas of strength and support!
DOWNLOAD FREEUse this quiz to check your grade 4 to 5 studentsβ understanding of number patterns. 10+ questions with answers covering a range of 4th and 5th grade number patterns topics to identify areas of strength and support!
DOWNLOAD FREEHere is some history on triangular numbers:
Triangular numbers are natural numbers. Carl Gauss and Pierre de Fermat are known for their work with number theory.
Triangular numbers were originally explored by the Pythagoreans who developed many relationships between different geometric shapes and numbers including triangular numbers, square numbers, pentagonal numbers (numbers represented within a regular pentagon) and hexagonal numbers (numbers represented within a regular hexagon).
These are sometimes known as polygonal numbers or figurate numbers.
Letβs take a look at another example of where triangular numbers show up β Pascalβs triangle.
Pascalβs triangle shows a pattern of numbers that represent the binomial coefficients of a binomial expansion (x+y)^{n} where the value in the row below is the sum of the two values above it.
You can identify the triangular numbers in Pascalβs triangle. They appear in Pascalβs triangle next to the natural numbers. This means that if we add these two values together, we will obtain the next triangular number below it.
Fun fact: Did you know: given two consecutive triangular numbers, the sum of triangular numbers is a square number!
How does this relate to high school math?
In order to calculate a triangular number, given a sequence:
Draw the next picture in the sequence, T_{4}.
Term 2 contains 3 squares.
Term 3 contains 6 squares.
6-3=3.2Add one more than this value to determine the next term.
Adding one to the previous difference means that the next term will have another row added below containing 3+1=4 squares.
3Continue until you reach the required term in the sequence.
Continue the pattern by drawing four squares in the last row.
The first four triangular numbers are 1, \, 3, \, 6, and 10. Is 21 a triangular number?
Calculate the difference between two consecutive terms.
You are given the first four triangular numbers, so you can calculate the difference between the last two terms, and add one more than this value to get the next number in the sequence.
10-6=4
Add one more than this value to determine the next term.
Adding one to the previous difference means that the next term will be 5 more than 10.
10+5=15
Continue until you reach the required term in the sequence.
The next term will be 6 more than 15.
15+6=21
So, 21 is a triangular number.
The first four triangular numbers are 1, \, 3, \, 6, and 10. What position in the sequence is the triangular number 45?
Calculate the difference between two consecutive terms.
You are given the first four triangular numbers, so you can calculate the difference between the last two terms and add one more than this value to get the next term.
10-6=4
Add one more than this value to determine the next term.
Adding one to the previous difference means that the next term will be 5 more than 10.
10+5=15
Continue until you reach the required term in the sequence.
The next term will be 6 more than 15.
15+6=21
The next term will be 7 more than 21.
21+7=28
The next term will be 8 more than 28.
28+8=36
The next term will be 9 more than 36.
36+9=45
So, 45 is the 9 th triangular number.
In order to calculate a triangular number, using the nth term:
Calculate the 10 th triangular number.
Identify the given information.
The 10 th triangular number means to find the number in the 10 th spot of the sequence or the 10 th term of the sequence, so, n=10.
Substitute the values given into the equation.
\hspace{3.5cm} T_n=\cfrac{1}{2} \, n(n+1)
Complete the calculation.
The 10 th term is 55.
Calculate the 50 th triangular number.
Identify the given information.
In this case, you are solving for the 50 th triangular number or the number in the 50 th position (50 th term), so, n=50.
Substitute the values given into the equation.
\hspace{3.5cm} T_n=\cfrac{1}{2} \, n(n+1)
Complete the calculation.
So, the 50 th term is 1275.
What term or position is the triangular number, 105, in?
Identify the given information.
In this case, the given information is the triangular number and you need to solve for the position or n.
T_n=105
Substitute the values given into the equation.
\hspace{3.5cm} T_n=\cfrac{1}{2} \, n(n+1)
Complete the calculation.
105=\cfrac{1}{2} \, n(n+1) \rightarrow Solving for n requires applying strategies to solve quadratic equations.
\begin{aligned}& 105=\cfrac{1}{2} \, n(n+1) \\\\ & 105=\cfrac{1}{2} \, n^2+\cfrac{1}{2} \, n \\\\ & 2\left(0=\cfrac{1}{2} \, n^2+\cfrac{1}{2} \, n-105\right) \\\\ & 0=n^2+n-210 \\\\ & (n+15)(n-14)=0 \\\\ & n+15=0 \quad n-14=0 \\\\ & n=- \, 15 \hspace{0.6cm} n=14 \end{aligned}
Since the position of a number in a sequence cannot be negative, 14 is the answer. This means that 105 is the 14 th triangular number or the number in the 14 th position.
1. Below is a pattern sequence.
Which picture shows the correct pattern for the term T_{5}?
The number of ovals increases by 1 for every new row above the triangle. The new row will contain 5 ovals, above the image for T_{4}.
Furthermore, the triangle in T_{5} will have 5 ovals for each side of the triangle.
2. What type of number is 28?
Square Number
Decimal Number
Cube Number
Triangular Number
The sequence of triangular numbers is
1, \, 3, \, 6, \, 10, \, 15, \, 21, \, 28, \, β¦
So, 28 is a triangular number.
3. What is the position of the term 36 in the sequence of triangular numbers?
9 th term
8 th term
36 th term
6 th term
The sequence of triangular numbers is
1, \, 3, \, 6, \, 10, \, 15, \, 21, \, 28, \, 36, \, β¦
The position of the value 36 is the 8 th term.
4. What is the value of the 12 th triangular number?
Substituting n=12 into the nth term for the triangular numbers sequence, you have
T_{12}=\cfrac{1}{2}\times{12}\times{13}=78.
5. What is the value of the 28 th triangular number?
Substituting n=28 into the nth term for the triangular numbers sequence, you have
T_{28}=\cfrac{1}{2}\times{28}\times{29}=406.
6. Which method correctly shows that 55 is a triangular number.
Forming an equation using the nth term of the sequence of triangular numbers, you have
\cfrac{1}{2} \, n(n+1)=55.
Multiplying both sides of the equation by 2, you have
n(n+1)=110.
Rearranging the equation to form a quadratic of the form
ax^{2}+bx+c=0 we have
n^{2}+n-110=0.
The quadratic factors to be,
(n-10)(n+11)=0
The two possible solutions are
n=10 or n=- \, 11.
The position in a sequence must be a positive integer, so n=10.
Therefore, 55 is the 10 th triangular number.
There are recursive formulas (also known as recurrence relation) and explicit formulas to find the number of a particular term of sequence. In algebra 2 and precalculus, you will learn how to create recursive formulas and explicit formulas for arithmetic and geometric sequences.
A factorial is an algorithm that takes any integer and multiplies it to all the preceding integers and itβs symbolized by β!β For example, 5!=5 \times 4 \times 3 \times 2 \times 1=120
Mersenne prime numbers are a sequence of numbers named after French theologian and mathematician Marin Mersenne. Mersenne prime numbers can be expressed as 2^{p}-1, where p is a prime number.
Mersenne primes, perfect numbers, and even perfect numbers are interlinked and studied at great length with theorems in a number theory course.
There are an infinite amount of sequences of numbers that include all types of numbers such as rational numbers (patterns between numerators and denominators), irrational numbers (patterns with square roots) and integers (patterns with positive and negative numbers).
Square triangular numbers are numbers that are both triangular numbers and perfect square numbers. These numbers can be represented by dots that form equilateral triangles and dots that form squares.
Yes, you can take the square root of any number. However, only square triangular numbers have whole numbers as square roots.
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