Understanding Triangular Numbers

A triangular number is a number which can be represented as a triangular grid of points. These numbers are formed by the sum of consecutive positive integers starting from 1.

The formula for the \(n\)-th triangular number, denoted as \(T_n\), is:

\( T_n = 1 + 2 + 3 + \dots + n = \frac{n(n+1)}{2} \)

Let's list the first few triangular numbers to understand the sequence:

• \( T_1 = 1 = \frac{1(1+1)}{2} \)
• \( T_2 = 1 + 2 = 3 = \frac{2(2+1)}{2} \)
• \( T_3 = 1 + 2 + 3 = 6 = \frac{3(3+1)}{2} \)
• \( T_4 = 1 + 2 + 3 + 4 = 10 = \frac{4(4+1)}{2} \)
• \( T_5 = 1 + 2 + 3 + 4 + 5 = 15 = \frac{5(5+1)}{2} \)
• \( T_6 = 1 + 2 + 3 + 4 + 5 + 6 = 21 = \frac{6(6+1)}{2} \)

Checking the Given Options for Triangular Numbers

We need to determine which of the given options is not a triangular number. We can do this by checking if each number can be expressed in the form \( \frac{n(n+1)}{2} \) for some positive integer \( n \).

Option 1: 15

Can 15 be a triangular number? We check if \( \frac{n(n+1)}{2} = 15 \).

\( n(n+1) = 15 \times 2 \)

\( n(n+1) = 30 \)

We are looking for a positive integer \(n\) such that \(n\) multiplied by \(n+1\) equals 30. We know that \(5 \times 6 = 30\). So, \(n=5\). Since we found an integer value for \(n\), 15 is a triangular number (\(T_5\)).

Option 2: 3

Can 3 be a triangular number? We check if \( \frac{n(n+1)}{2} = 3 \).

\( n(n+1) = 3 \times 2 \)

\( n(n+1) = 6 \)

We are looking for a positive integer \(n\) such that \(n\) multiplied by \(n+1\) equals 6. We know that \(2 \times 3 = 6\). So, \(n=2\). Since we found an integer value for \(n\), 3 is a triangular number (\(T_2\)).

Option 3: 10

Can 10 be a triangular number? We check if \( \frac{n(n+1)}{2} = 10 \).

\( n(n+1) = 10 \times 2 \)

\( n(n+1) = 20 \)

We are looking for a positive integer \(n\) such that \(n\) multiplied by \(n+1\) equals 20. We know that \(4 \times 5 = 20\). So, \(n=4\). Since we found an integer value for \(n\), 10 is a triangular number (\(T_4\)).

Option 4: 5

Can 5 be a triangular number? We check if \( \frac{n(n+1)}{2} = 5 \).

\( n(n+1) = 5 \times 2 \)

\( n(n+1) = 10 \)

We are looking for a positive integer \(n\) such that \(n\) multiplied by \(n+1\) equals 10. Let's check integer values for \(n\):

• If \( n=1 \), \( n(n+1) = 1(2) = 2 \)
• If \( n=2 \), \( n(n+1) = 2(3) = 6 \)
• If \( n=3 \), \( n(n+1) = 3(4) = 12 \)

As \(n\) increases, \(n(n+1)\) also increases. We can see that 10 falls between 6 (for \(n=2\)) and 12 (for \(n=3\)). There is no positive integer \(n\) for which \(n(n+1) = 10\). Therefore, 5 is not a triangular number.

Identifying the Non-Triangular Number

Based on our analysis, the numbers 15, 3, and 10 are triangular numbers. The number 5 cannot be expressed in the form \( \frac{n(n+1)}{2} \) for any positive integer \(n\).

Thus, 5 is not a triangular number.

Revision Table: Triangular Numbers

Additional Information on Triangular Numbers

Triangular numbers have many interesting properties. For example:

• The sum of two consecutive triangular numbers is always a perfect square. \( T_n + T_{n+1} = \frac{n(n+1)}{2} + \frac{(n+1)(n+2)}{2} = \frac{(n+1)(n + n+2)}{2} = \frac{(n+1)(2n+2)}{2} = \frac{(n+1)2(n+1)}{2} = (n+1)^2 \).
• Every perfect number is a triangular number (except for 6, which is \(T_3\)). For example, 28 is a perfect number (\(1+2+4+7+14=28\)) and it is also a triangular number (\(T_7 = 28\)).
• A positive integer \(x\) is a triangular number if and only if \(8x + 1\) is a perfect square. Let's check this for the options:
  • For 15: \(8(15) + 1 = 120 + 1 = 121\). \(121 = 11^2\). Yes, 15 is triangular.
  • For 3: \(8(3) + 1 = 24 + 1 = 25\). \(25 = 5^2\). Yes, 3 is triangular.
  • For 10: \(8(10) + 1 = 80 + 1 = 81\). \(81 = 9^2\). Yes, 10 is triangular.
  • For 5: \(8(5) + 1 = 40 + 1 = 41\). 41 is not a perfect square. No, 5 is not triangular.

This property confirms that 5 is not a triangular number, while 3, 10, and 15 are.