Each term of the triangular number sequence is the sum of the natural numbers up to that term. For example, 15, the fifth triangular number, is equal to 1 + 2 + 3 + 4 + 5. This is represented by the diagram in the figure, where each triangular number is represented by a triangular array of dots. Can you draw the patterns for the next two terms of the sequence?
Step-by-step solution
Idea: The nth triangular pattern has n rows, with 1 dot in the top row, 2 in the next, and so on. To get the next pattern, add one more row with one more dot than the last row.
- The 5th pattern has 5 rows (1, 2, 3, 4, 5 dots) = 15 dots.
- 6th pattern: add a bottom row of 6 dots. Dots = 15 + 6 = 21.1 mark
- 7th pattern: add a bottom row of 7 dots. Dots = 21 + 7 = 28. The drawings are shown below.1 mark
Check: 1 + 2 + 3 + 4 + 5 + 6 = 21 ✓ and 21 + 7 = 28 ✓.
Answer to write in the exam
5th pattern: rows of 1, 2, 3, 4, 5 dots = 15 dots
6th pattern: add a row of 6 dots → 15 + 6 = 21 dots
7th pattern: add a row of 7 dots → 21 + 7 = 28 dots
∴ Next two patterns: triangles of 21 dots (6 rows) and 28 dots (7 rows)
Common mistakes that cost marks
- Adding 5 again (15 + 5 = 20). The new row always has one more dot than the previous bottom row: 6, then 7.
- Drawing the new rows with gaps or uneven spacing, so the shape is no longer a triangle. Each row should be centred under the one above.
- Writing 36 as the next triangular number. 36 = 1 + 2 + … + 8 is the 8th term, not the 6th.
How this can come in the exam
How many dots are in the bottom row of the triangular dot pattern for 66?
- 10
- 11
- 12
- 66
Show answer
(B) 11
1 + 2 + … + 11 = 66, so 11 rows; the bottom row has 11 dots.
Try one yourself
How many dots are needed to draw the 10th triangular pattern? How many dots are in its bottom row?
Show answer
1 + 2 + … + 10 = 55 dots; the bottom row has 10 dots.
More questions like this
- 1 = 1, 4 = 1 + 3, 9 = 1 + 3 + 5, 16 = 1 + 3 + 5 + 7, and so on. Each term in the square number sequence is the sum of the odd numbers up to that term. This interesting relationship between the odd numbers and square numbers can be represented by the diagram in the figure. Can you explain the relationship?
- Consider the sequence 1, 4, 7, 10, 13, … Can you predict the next four terms? Can you derive the first 10 terms of the sequence obtained by adding all the terms up to a given term of this sequence? (Hint: The first term is 1. The second term is 1 + 4 = 5, the third term is 1 + 4 + 7 = 12, and so on.)
- Can you write t5, t6, t7 and t8 for the sequence of triangular numbers?
- Can you think of any other kinds of sequences? List out five different types of sequences and discuss their properties with your friends.
- Consider the expression un = 2n − 1. This states that the nth term of the sequence is given by the rule 2n − 1.
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