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. 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? 1 st 2 nd 3 rd 4 th

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 The 12th triangular number is 78. The 24th triangular number is 300. Which triangular number is 120? becomes on simplifying n 2 – n – 240 = 0 (n – 15)(n + 16) = 0 n = 15, n = -16. Therefore n = 15 is the solution.

Number of terms Sum 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