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Triangular Numbers Trivia Quiz

I have created quizzes about Fibonacci numbers, prime numbers and perfect numbers. Here is my latest installment- introducing the amazing and mysterious triangular numbers! Enjoy and thanks for playing.

A multiple-choice quiz by Matthew_07. Estimated time: 4 mins.

Author
Matthew_07

Time
4 mins

Type
Multiple Choice

Quiz #
287,349

Updated
Dec 03 21

# Qns
10

Difficulty
Average

Avg Score
7 / 10

Plays
898

One at a Time - Single Page - Timed Game

Question 1 of 10

1. The first few triangular numbers are 1, 3, 6, 10, 15, 21... What is the general formula for finding the nth triangular number? Hint

n - (n-1) - (n-2) - ... - 2 - 1

n + (n-1) + (n-2) + ... + 2 + 1

n! + (n-1)! + (n-2)! + ... + 2! + 1!

n! - (n-1)! - (n-2)! - ... - 2! - 1!

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Question 2 of 10

2. Which of the following formulas provides an alternative way to find the nth triangular number? Hint

(n^3 + n)/2

(n^2 + n)/3

(n^3 + n)/3

(n^2 + n)/2

3. According to www.mathworld.wolfram.com, a figural number is a number that can be represented by a regular geometrical arrangement of equally spaced points. So, can triangular numbers be categorized as figural numbers?

Yes

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Question 4 of 10

4. Given the first few terms of triangular numbers: 1, 3, 6, 10, 15, 21..., the sum of any 2 consecutive triangular numbers is a (an) _____ number. Hint

Prime

Square

Odd

Perfect

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Question 5 of 10

5. Here are the first 8 triangular numbers: 1, 3, 6, 10, 15, 21, 28, 36. Notice that the first 2 terms are odd, followed by another 2 even terms, and then another 2 odd terms, before 2 even terms again. In general, does this pattern hold true for all triangular numbers?

Yes

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Question 6 of 10

6. The sum of all the reciprocals of triangular numbers (namely 1/1 + 1/3 + 1/6 + 1/10 + 1/15 + 1/21 +...) is ___. Hint

NEXT>

Question 7 of 10

7. The smallest palindromic triangular number is 55. Is the Number of the Beast, 666 another palindromic triangular number?

Yes

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Question 8 of 10

8. We can use triangular numbers to solve mathematical problems. Suppose that 5 friends met in a birthday party and they wanted to shake each other's hands. So, how many handshakes would there be all together? Hint

4 + 3 + 2 + 1 = 10

5 + 4 + 3 + 2 + 1 = 15

4! + 3! + 2! + 1! = 33

5! + 4! + 3! + 2! + 1! = 153

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Question 9 of 10

9. Which mathematician proved that every positive integer can be represented by the sum of, at most, 3 triangular numbers? Hint

Newton

Gauss

Fermat

Cauchy

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Question 10 of 10

10. A very popular method to test whether a given number is a triangular number or not is by substituting that number into the formula n = 0.5 x [ [square root of (8t + 1)] -1 ], where t is the given number. If n is a whole number (1, 2, 3...), then the number is a triangular number.

Yes

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Quiz Answer Key and Fun Facts

1. The first few triangular numbers are 1, 3, 6, 10, 15, 21... What is the general formula for finding the nth triangular number?

Answer: n + (n-1) + (n-2) + ... + 2 + 1

The general formula for triangular numbers is n + (n-1) + (n-2) + ... + 2 + 1. Let say we want to find the 5th triangular number. So we just need to substitute n with 5 to get our answer, which is 5 + 4 + 3 + 2 + 1 = 15.

2. Which of the following formulas provides an alternative way to find the nth triangular number?

Answer: (n^2 + n)/2

From n + (n-1) + (n-2) + ... + 2 + 1, we can do some mathematical manipulation to arrive at another formula, which is (n^2 + n)/2.

Answer: Yes

Figural numbers are also known as figurate numbers. Other examples of figural numbers are such as tetrahedral numbers and pentatopic numbers.

4. Given the first few terms of triangular numbers: 1, 3, 6, 10, 15, 21..., the sum of any 2 consecutive triangular numbers is a (an) _____ number.

Answer: Square

Notice that 1 + 3 = 4, 3 + 6 = 9, 6 + 10 = 16, 10 + 15 = 25, 15 + 21 = 36 are all square numbers.

Answer: Yes

We can prove this by mathematical induction. This is one of the interesting properties of triangular numbers.

1 (odd)
1 (odd) + 2 (even) = 3 (odd)
1 (odd) + 2 (even) + 3 (odd) = 6 (even)
1 (odd) + 2 (even) + 3 (odd) + 4 (even) = 10 (even)

6. The sum of all the reciprocals of triangular numbers (namely 1/1 + 1/3 + 1/6 + 1/10 + 1/15 + 1/21 +...) is ___.

Answer: 2

Since the first term of the summation is 1, then the answer must be greater than 1. The sum of all the reciprocals of triangular numbers can be found easily from the telescoping series 2{1/1 + 1/3 + 1/6 + 1/10 + ... + 1/[n(n+1)] + ...}

7. The smallest palindromic triangular number is 55. Is the Number of the Beast, 666 another palindromic triangular number?

Answer: Yes

Yes, 666 is the fifth palindromic triangular number, after 55, 66, 171 and 595. Notice that 666 is written as DCLXVI in Roman numerals.

Answer: 4 + 3 + 2 + 1 = 10

In general, if there are n people, then there will be (n-1) + (n-2) + ... + 2 + 1 handshakes.

Let's check our answer. In the case where there are 5 people (denote them as A, B, C, D and E), the 10 combinations are AB, AC, AD, AE, BC, BD, BE, CD, CE and DE.

9. Which mathematician proved that every positive integer can be represented by the sum of, at most, 3 triangular numbers?

Answer: Gauss

This was first proposed by Fermat and it was called Fermat's Polygonal Number Theorem.

Carl Friedrich Gauss was a German mathematician. He found out this property and wrote down "Heureka! num = triangle + triangle + triangle."

Answer: Yes

Furthermore, if n is a whole number, then, the number t is the nth triangular number. For example, we notice that when we substitute t = 6 into the formula, we will get n = 3. So, we can make our conclusion that 6 is the third triangular number.

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I hope you enjoy my quiz and learn something new as well. Thanks for playing and have a nice day!

Source: Author Matthew_07

This quiz was reviewed by FunTrivia editor crisw before going online.
Any errors found in FunTrivia content are routinely corrected through our feedback system.

Related Quizzes

This quiz is part of series Fun with Numbers:

A collection of my mathematics quizzes, covering different types of numbers with interesting properties. This list includes composite numbers, consecutive numbers, Fibonacci numbers, palindromic numbers, perfect numbers, prime numbers, square numbers, and triangular numbers.

  1. Composite Numbers Average
  2. Consecutive Numbers Average
  3. Fibonacci Numbers Average
  4. Palindromic Numbers Average
  5. Perfect Numbers Average
  6. Prime Numbers Tough
  7. Square Numbers Average
  8. Triangular Numbers Average

Referenced Topics

General People Science Math Specific Math Topics German Language Roman Numerals Mathematicians Theorems Germans: Famous & Historical

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