Interesting Triangular Number Patterns: From Nrich
Another pattern: The sum of two consecutive triangular numbers is a square number.
What are triangular numbers? Let's exam the first 4 triangular numbers:
The 1st number is "1".
The 2nd number is "3" (1 + 2)
The 3rd number is "6" (1 + 2 + 3)
The 4th number is "10" (1 + 2 + 3 + 4)
.
.
The nth number is \(\frac{n(n+1)}{2}\)
It's the same as finding out the sum of the first "n" natural numbers.
Let's look at this question based on the song "On the Twelve Day of Christmas"
(You can listen to this on Youtube,)
On the Twelve Day of Christmas
On the first day of Christmas
my true love gave to me
a Partridge in a Pear Tree
On the second day of Christmas,
My true love gave to me,
Two Turtle Doves,
And a Partridge in a Pear Tree.
On the third day of Christmas,
My true love gave to me,
Three French Hens,
Two Turtle Doves,
And a Partridge in a Pear Tree
On the fourth day of Christmas,
My true love gave to me,
Four Calling Birds,
Three French Hens,
Two Turtle Doves,
And a Partridge in a Pear Tree.
The question is "How many gifts were given out on the Day of Christmas?"
Solution I"
1st Day: 1
2nd Day: 1 + 2
3rd Day 1 + 2 + 3
4th Day 1 + 2 + 3 + 4
.
.
12th Day 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12
Altogether, you'll have 12 * 1 + 11 * 2 + 10 * 3 + 9 * 4 + 8 * 5 + 7 * 6 + 6 * 7 + 5 * 8 + 4 * 9 + 3 * 10 + 2 * 11 + 1 * 12 = 12 + 22 + 30 + 36 + 40 + 42 + 42 + 40 + 36 + 30 + 22 + 12 = 364
Solution II: The sum of the "n" triangular number is a tetrahedral number.
To get the sum, you use \(\frac{n(n+1)(n+2)}{6}\)
n = 12 and \(\frac{12 (13)(14)}{6}= 364 \)
Here is a proof without words.
Applicable questions: (Answers and solutions below)
#1 Some numbers are both triangular as well as square numbers. What is the sum of the first three positive numbers that are both triangular numbers and square numbers?
#2 What is the 10th triangular number? What is the sum of the first 10 triangular numbers?
#3 What is the 20th triangular number?
#4 One chord can divide a circle into at most 2 regions, Two chords can divide a circle at most into 4 regions. Three chords can divide a circle into at most seven regions. What is the maximum number of regions that a circle can be divided into by 50 chords?
#5: Following the pattern, how many triangles are there in the 15th image?
Answers:
#1 1262 The first 3 positive square triangular numbers are: 1, 36 (n = 8) and 1225 (n = 49).
#2 55; 220 a. \(\frac{10*11}{2}=55\) b. \(\frac{10*11*12}{6}=220\)
#3 210 \(\frac{20*21}{2}=210\)
#4 1276
1 chord : 2 regions or \(\boxed{1}\) + 1
2 chords: 4 regions or \(\boxed{1}\) + 1 + 2
3 chords: 7 regions or \(\boxed{1}\) + 1 + 2 + 3
.
.
50 chords:\(\boxed{1}\) + 1 + 2 + 3 + ...+ 50 = 1 + \(\frac{50*51}{2}\) =1276
#5: 120
The first image has just one triangle, The second three triangles. The third 6 triangles total.
It follows the triangular number pattern. The 15th triangular is \(\frac{15*16}{2}\) =120
