In the number series 0, 3, 8, 15, 24, ?, determine the next term by recognizing the pattern related to squares of natural numbers.

Introduction / Context:
This question presents a sequence of numbers that gradually increases: 0, 3, 8, 15, 24, ?. The task is to find the next term in the series. These numbers look very close to perfect squares, which is a strong clue that the series might be constructed using simple square numbers of consecutive integers.

Given Data / Assumptions:
The terms given are:

• 0
• 3
• 8
• 15
• 24
• ?

We assume a simple and consistent rule, very likely involving n^2 for small integer values of n, adjusted by a fixed quantity.

Concept / Approach:
A good way to approach this kind of problem is to compare each term with square numbers of 1, 2, 3, 4, 5, 6, and so on. If the difference between each term and the square shows a consistent pattern, we can use that pattern to find the next number. Here, the pattern turns out to be n^2 - 1.

Step-by-Step Solution:
Step 1: Check the first term. For n = 1, n^2 - 1 = 1^2 - 1 = 0, which matches the first term.Step 2: Check the second term. For n = 2, n^2 - 1 = 2^2 - 1 = 4 - 1 = 3, which matches the second term.Step 3: Check the third term. For n = 3, n^2 - 1 = 9 - 1 = 8, matching the third term.Step 4: Check the fourth term. For n = 4, n^2 - 1 = 16 - 1 = 15, matching the fourth term.Step 5: Check the fifth term. For n = 5, n^2 - 1 = 25 - 1 = 24, matching the fifth term.Step 6: For the sixth position, we take n = 6. Then n^2 - 1 = 36 - 1 = 35. This should be the next term.

Verification / Alternative check:
The general term of the sequence can be written as a(n) = n^2 - 1 for n = 1, 2, 3, 4, 5, 6. Substituting n = 1 to 5 reproduces all the given terms exactly, so extending it to n = 6 gives a(6) = 35 without any contradiction. Because the pattern is simple and exact, this strongly confirms our answer.

Why Other Options Are Wrong:
Values like 34, 36 and 37 do not equal n^2 - 1 for n = 6. Specifically, 36 is equal to 6^2, 34 is 36 - 2, and 37 is 36 + 1. Choosing any of them would break the simple and elegant square minus one pattern that fits all previous terms perfectly.

Common Pitfalls:
Some candidates may look only at the differences 3, 5, 7, 9 and decide the next difference should be 11, giving 24 + 11 = 35. Interestingly, this reasoning also leads to 35, because the differences form consecutive odd numbers. However, recognizing the deeper structure as n^2 - 1 helps confirm the result more confidently and is useful in similar problems involving square numbers.

Final Answer:
The next number in the series, following the pattern a(n) = n^2 - 1, is 35.