Triangular Numbers

1
Triangular Numbers

• An Investigation

2

• Triangular Numbers
• Triangular numbers are made by forming triangular
  patterns with counters. The first four triangular
  numbers are shown in the diagram.
• 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?

3

• 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?

4

• 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
• The 12th triangular number is 78.
• The 24th triangular number is 300.
• Which triangular number is 120?
• becomes on simplifying n2 n 240 0
• (n 15)(n 16) 0
• n 15 , n -16.
• Therefore n 15 is the solution.

5
Forming the sums
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
So the sum of the first 15 triangular numbers is
given by