Step 1: Understanding the Concept:
Number series puzzles test your ability to recognize logical patterns and mathematical relationships between consecutive terms.
The objective is to identify a "rule" that consistently transforms one term into the next.
In this sequence (\(5, 11, 23, 47\)), the numbers are increasing quite rapidly, which suggests that the rule involves multiplication or an expanding addition pattern.
We can analyze the series by looking at two perspectives: a recursive formula (using the previous term) or a difference-based pattern (looking at the gaps between numbers).
Step 2: Key Formula or Approach:
We can test the relationship:
\[ \text{Term}_{n+1} = (\text{Term}_n \times 2) + 1 \]
Alternatively, we can examine the series of differences:
\[ \text{Difference} = \text{Term}_{n+1} - \text{Term}_n \]
and check if those differences follow a geometric or arithmetic progression.
Step 3: Detailed Explanation:
Let us apply the multiplication rule first to see if it is consistent across the given terms.
- From Term 1 to Term 2: \((5 \times 2) + 1 = 10 + 1 = 11\). (Pattern holds)
- From Term 2 to Term 3: \((11 \times 2) + 1 = 22 + 1 = 23\). (Pattern holds)
- From Term 3 to Term 4: \((23 \times 2) + 1 = 46 + 1 = 47\). (Pattern holds)
Since the logic is consistent, we apply it to the fourth term to find the missing fifth term:
- \(\text{Missing Term} = (47 \times 2) + 1\)
- \(\text{Missing Term} = 94 + 1 = 95\).
Let's double-check this using the difference method:
- Difference between \(11\) and \(5 = 11 - 5 = 6\)
- Difference between \(23\) and \(11 = 23 - 11 = 12\)
- Difference between \(47\) and \(23 = 47 - 23 = 24\)
The sequence of differences is \(6, 12, 24, \dots\)
Notice that each difference is exactly twice the previous difference.
Following this logic, the next difference must be \(24 \times 2 = 48\).
- \(\text{Missing Term} = \text{Previous Term} + \text{Next Difference}\)
- \(\text{Missing Term} = 47 + 48 = 95\).
Both mathematical paths lead to the same result, confirming that \(95\) is the logically correct missing number.
Step 4: Final Answer:
The missing number in the series is 95.