Which of the following numbers is not a triangular number?

Introduction / Context:
This question checks understanding of triangular numbers. A triangular number counts objects that can be arranged in an equilateral triangular pattern. Triangular numbers appear in many combinatorial and geometric contexts. Knowing how to recognise and test triangular numbers is useful in number theory and aptitude problems.

Given Data / Assumptions:

• We are given the options 15, 10, 5, 3, and 21.
• We must determine which one is not a triangular number.
• A triangular number Tn is given by Tn = n * (n + 1) / 2 for some positive integer n.
• Numbers are positive integers.

Concept / Approach:
To test if a number N is triangular, we can either list triangular numbers or invert the formula Tn = n * (n + 1) / 2. If N is triangular, the equation n^2 + n - 2N = 0 will have a positive integer solution for n. Equivalently, 8N + 1 must be a perfect square, because the discriminant of the quadratic in n is 1 + 8N. We can use either the direct listing or the discriminant method to check each option.

Step-by-Step Solution:
Step 1: Recall the formula for triangular numbers: Tn = n * (n + 1) / 2.Step 2: List the first few triangular numbers: T1 = 1, T2 = 3, T3 = 6, T4 = 10, T5 = 15, T6 = 21, and so on.Step 3: Compare each option with this list.Step 4: 3 appears as T2, so 3 is triangular.Step 5: 10 appears as T4, so 10 is triangular.Step 6: 15 appears as T5, so 15 is triangular.Step 7: 21 appears as T6, so 21 is also triangular.Step 8: The number 5 does not appear among these triangular numbers, and it is not formed by n * (n + 1) / 2 for any small integer n.Step 9: Therefore, 5 is not a triangular number.

Verification / Alternative check:
Use the discriminant test. For a number N to be triangular, 8N + 1 should be a perfect square. For N = 5, compute 8 * 5 + 1 = 41, which is not a perfect square. For N = 3, 8 * 3 + 1 = 25 = 5^2, so 3 is triangular. For N = 10, 8 * 10 + 1 = 81 = 9^2, so 10 is triangular. For N = 15, 8 * 15 + 1 = 121 = 11^2, so 15 is triangular. For N = 21, 8 * 21 + 1 = 169 = 13^2, so 21 is triangular. This confirms that 5 alone fails the triangular test.

Why Other Options Are Wrong:
Option 15: It is T5, a known triangular number.
Option 10: It is T4, clearly triangular.
Option 3: It is T2, the second triangular number.
Option 21: It is T6, also triangular.

Common Pitfalls:
Students sometimes mistakenly think that any number that can form some geometric pattern is triangular, without checking the formula. Others might confuse triangular numbers with square numbers or other figurate numbers. Using the straightforward formula Tn = n * (n + 1) / 2, or the discriminant test 8N + 1 being a perfect square, helps avoid such confusion and gives a quick, reliable check.

Final Answer:
The number that is not a triangular number is 5.