ریاضی
What Is a Triangular Number?
1, 3, 6, 10, 15 — the running totals of the counting numbers. Where the n(n+1)/2 formula comes from, and why handshake questions give the same answers.
Triangular number
A triangular number is the total you get by adding the whole numbers from 1 up to some point, so that many dots can be arranged in an equal-sided triangle.
- دیگر نام
- Triangle number
- طلبہ کہاں پڑھتے ہیں
- Grade 5 or 6 mathematics as a dot pattern, and again in GCSE and IGCSE sequence questions that ask for the nth term of 1, 3, 6, 10.
مختصر جواب
A triangular number is a running total of the counting numbers: 1, 3, 6, 10, 15, 21. The nth one equals n(n+1)/2, so the tenth is 10 × 11 ÷ 2 = 55. The name comes from the fact that many dots stacking into an equal-sided triangle.
ایک مثال
1, 3, 6, 10, 15, 21, 28, 36, 45, 55
Each number is the one before it plus the next counting number: 10 + 5 = 15, then 15 + 6 = 21. The tenth term can be found without listing the first nine, because the formula n(n+1)/2 gives 10 × 11 ÷ 2 = 55 directly.
Where the triangle comes in
Put down one dot. Underneath it put a row of two, then a row of three, and so on. After each new row you have a triangle, and the number of dots in it is a triangular number: 1, then 3, then 6, then 10.
That is the whole idea, and it explains the gaps. The jumps between consecutive triangular numbers go 2, 3, 4, 5, 6, because each new row is one dot longer than the last. The differences between those differences are all 1, which is what makes the sequence quadratic rather than linear.
Why the formula is n(n+1)/2
Take the triangle of 10 dots — four rows of 1, 2, 3, 4 — and place an identical triangle upside down beside it. The two together make a solid rectangle 4 dots wide and 5 dots tall, which holds 20 dots. One triangle is half of that, so 10.
The same trick works for any n: two copies of the nth triangle make a rectangle n by (n + 1), so one triangle is n(n+1)/2. Written as a sequence rule, that is ½n² + ½n, which is why these appear in questions about quadratic sequences.
The questions that are secretly about triangular numbers
Exam questions rarely use the words. They describe a situation where every item is paired with every other item, and that count is always triangular.
In a room of 11 people where everyone shakes hands once with everyone else, the number of handshakes is 10 + 9 + 8 + … + 1 = 55 — the tenth triangular number. The pattern is n(n−1)/2 for n people, which is the triangular formula shifted down by one because nobody shakes their own hand.
- Handshakes among n people: n(n−1)/2
- Straight lines joining n points, no three in a line: n(n−1)/2
- Matches in a round-robin league where every team plays every other team once
- Skittles or snooker-style triangular stacks: 10 pins is the fourth triangular number, 15 reds is the fifth
- The third diagonal of Pascal's triangle: 1, 3, 6, 10, 15
- The tenth triangular number, and the number of handshakes in a room of eleven people
- 55The tenth triangular number, and the number of handshakes in a room of eleven people
عام سوالات
Is 1 a triangular number?
Yes. One dot is the smallest triangle the pattern allows, and putting n = 1 into n(n+1)/2 gives 1 × 2 ÷ 2 = 1. Sequences usually start there. Whether 0 counts is a matter of convention; school questions begin the list at 1.
Can a number be both triangular and square?
Yes, though it is rare. 36 is the eighth triangular number and also 6 squared. 1225 is the forty-ninth triangular number and 35 squared. These are called square triangular numbers, and they get very far apart very quickly — the next one is over a million.
How do I find the nth term of 1, 3, 6, 10, 15?
The first differences are 2, 3, 4, 5 and the second differences are all 1, which tells you the rule contains ½n². Subtracting ½n² from each term leaves ½, 1, 1½, 2, which is ½n. So the nth term is ½n² + ½n, the same thing as n(n+1)/2.
What is the difference between triangular and square numbers?
Square numbers are the totals of dots in a square: 1, 4, 9, 16. Triangular numbers are the totals in a triangle: 1, 3, 6, 10. Add two consecutive triangular numbers and you always get a square — 6 + 10 = 16 — because two triangles fit together into a square.
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