Concept 4 of 19Foundation
Watch on YouTubeVideoSquare, triangular, cube numbers
Three families you'll meet constantly:
- Square numbers (n²): 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 …
- Triangular numbers (Tₙ = n(n+1)/2): 1, 3, 6, 10, 15, 21, 28, 36, 45, 55 …
- Cube numbers (n³): 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000 …
Beautiful link: the sum of the first n odd numbers = n². So 1 = 1², 1+3 = 2², 1+3+5 = 3²…
Example
The 10th triangular number = 10·11/2 = 55.
Is 120 triangular? 15·16/2 = 120 ✓. Yes, it's T₁₅.
Is 120 triangular? 15·16/2 = 120 ✓. Yes, it's T₁₅.
💡 Tip:Memorize the first 10 squares (up to 100) and triangular numbers (up to 55). Saves seconds on every olympiad.
▸Why does this work? (derivation)
Why Tₙ = n(n+1)/2? A triangular arrangement of n rows has 1 + 2 + … + n dots. Pair first with last, second with second-last: each pair sums to (n+1), and there are n/2 pairs → total n(n+1)/2.
🎯 Try it!
5 questions to check what you just read.
0 / 5
- Q1.10th square number:
- Q2.Is 45 a triangular number?
- Q3.6th cube number:
- Q4.Sum 1 + 3 + 5 + 7 + 9 = ?
- Q5.Which is a perfect cube?