Step 1: Understanding the Concept:
A number series is a sequence of numbers structured according to a logical mathematical rule. To find the missing term, you need to identify the mathematical progression pattern by analyzing the relationships between consecutive numbers. This can be done by looking at their differences, ratios, or exponential powers.
Step 2: Key Formula or Approach:
Let's evaluate this sequence using two different common approaches: finding the differences between successive numbers, and checking for perfect square adjustments.
Step 3: Detailed Explanation:
Method 1: Analysis of Differences
Let's find the step gap between each consecutive number in the given sequence:
- $3 \to 8 \implies 8 - 3 = \mathbf{+5}$
- $8 \to 15 \implies 15 - 8 = \mathbf{+7}$
- $15 \to 24 \implies 24 - 15 = \mathbf{+9}$
- $24 \to 35 \implies 35 - 24 = \mathbf{+11}$
Notice the trend in the calculated differences: 5, 7, 9, 11. The gaps form a secondary arithmetic sequence of consecutive odd numbers increasing by $+2$ at each step. Following this established rule, the next difference in the pattern must be:
$$ 11 + 2 = \mathbf{+13} $$
Now, add this difference to the last known number in our sequence to calculate the missing value:
$$ 35 + 13 = \mathbf{48} $$
Method 2: Perfect Squares Pattern ($n^2 - 1$)
Let's check how each term relates to the nearest perfect square number:
- $2^2 - 1 = 4 - 1 = 3$
- $3^2 - 1 = 9 - 1 = 8$
- $4^2 - 1 = 16 - 1 = 15$
- $5^2 - 1 = 25 - 1 = 24$
- $6^2 - 1 = 36 - 1 = 35$
The terms track the formula $n^2 - 1$ for sequential integers starting from $n=2$. Following this rule, the next term in the progression must be based on $n=7$:
$$ 7^2 - 1 = 49 - 1 = \mathbf{48} $$
Both independent methods confirm that the next number is 48. This matches option (B).
Step 4: Final Answer:
The next number in the series is 48.