Download Standard Grade Mathematics Investigations

Triangular Numbers
An Investigation
Triangular Numbers
Triangular numbers are made by forming triangular patterns with
counters. The first four triangular numbers are shown in the
diagram.
1st
2nd
3rd
4th
The first triangular number is made with just one counter and so
is one. The second triangular number is 3. The 3rd triangular
number is 6 and the 4th triangular number is 10.
Write down the next 4 triangular numbers.
What is the 12th triangular number?
What is the 24th triangular number?
Which triangular number is equal to 120?
Investigate the sequence obtained when you add together
The first two triangular numbers.
The first three triangular numbers.
The first four triangular numbers.
Can you find a formula for the sum of the first n triangular
numbers?
What is the sum of the first 15 triangular numbers?
Solution
The next 4 triangular numbers are 15, 21, 28 and 36.
Pupils could be asked to obtain a formula to generate triangular numbers
ie
1
n(n  1)
2
The 12th triangular number is 78.
The 24th triangular number is 300.
Which triangular number is 120?
1
n(n  1)  120
2
becomes on simplifying n2 – n – 240 = 0
(n – 15)(n + 16) = 0
n = 15 , n = -16.
Therefore n = 15 is the solution.
Forming the sums
Number of terms
1
2
3
4
5
6
Sum
1
4
10
20
35
56
If a difference table is formed then third differences
are constant so a cubic polynomial will fit the data.
This turns out to be the cubic
n
(n  1)(n  2)
6
So the sum of the first 15 triangular numbers is given by
15
(16)(17 )  680
6