Step 1: Understanding the Concept:
Number series puzzles require identifying a consistent logical or mathematical rule that governs the progression of numbers.
Standard patterns include constant differences, increasing differences, geometric ratios, or relationships to perfect squares and cubes.
Step 2: Key Formula or Approach:
We can evaluate this series using two distinct methodologies:
Method 1: Analyzing the "Difference of Differences" or simply the arithmetic gaps.
Method 2: Recognizing the relationship of each term to perfect squares (Square - Constant).
Step 3: Detailed Explanation:
Approach 1: Finding Arithmetic Differences
Let's find the difference between consecutive terms:
\[ 8 - 3 = +5 \]
\[ 15 - 8 = +7 \]
\[ 24 - 15 = +9 \]
\[ 35 - 24 = +11 \]
The differences are 5, 7, 9, 11.
Observe that these are consecutive odd numbers starting from 5.
The next odd number in this sequence must be 13.
To find the missing term, we add this difference to the last known term:
\[ \text{Missing Number} = 35 + 13 = 48 \]
Approach 2: Perfect Square Relationship
Notice that each number in the series is very close to a perfect square:
\[ 2^2 - 1 = 4 - 1 = 3 \]
\[ 3^2 - 1 = 9 - 1 = 8 \]
\[ 4^2 - 1 = 16 - 1 = 15 \]
\[ 5^2 - 1 = 25 - 1 = 24 \]
\[ 6^2 - 1 = 36 - 1 = 35 \]
Following this pattern (\(n^2 - 1\)), the next term corresponds to \(n = 7\):
\[ 7^2 - 1 = 49 - 1 = 48 \]
Both logical paths lead to the same result, reinforcing its validity.
Step 4: Final Answer:
The correct missing number is 48.