Finding the Pattern in the Number Series

The given number series is: 5, 5, 11, 35, 95, 215, ?

To find the next number in this series, we need to identify the underlying pattern. A common approach for such problems is to examine the differences between consecutive terms.

Calculating Differences Between Terms

Let's calculate the first level of differences between successive terms:

• Difference between the 2nd term (5) and the 1st term (5): \(5 - 5 = 0\)
• Difference between the 3rd term (11) and the 2nd term (5): \(11 - 5 = 6\)
• Difference between the 4th term (35) and the 3rd term (11): \(35 - 11 = 24\)
• Difference between the 5th term (95) and the 4th term (35): \(95 - 35 = 60\)
• Difference between the 6th term (215) and the 5th term (95): \(215 - 95 = 120\)

The sequence of these first differences is: 0, 6, 24, 60, 120.

Analyzing Second Differences

Let's calculate the differences between these first differences:

• Difference between the 2nd first difference (6) and the 1st (0): \(6 - 0 = 6\)
• Difference between the 3rd first difference (24) and the 2nd (6): \(24 - 6 = 18\)
• Difference between the 4th first difference (60) and the 3rd (24): \(60 - 24 = 36\)
• Difference between the 5th first difference (120) and the 4th (60): \(120 - 60 = 60\)

The sequence of second differences is: 6, 18, 36, 60.

Analyzing Third Differences

Let's calculate the differences between these second differences:

• Difference between the 2nd second difference (18) and the 1st (6): \(18 - 6 = 12\)
• Difference between the 3rd second difference (36) and the 2nd (18): \(36 - 18 = 18\)
• Difference between the 4th second difference (60) and the 3rd (36): \(60 - 36 = 24\)

The sequence of third differences is: 12, 18, 24.

Identifying the Constant Difference

Looking at the third differences (12, 18, 24), we can see a constant difference between consecutive terms: \(18 - 12 = 6\) and \(24 - 18 = 6\). This indicates that the next third difference will also follow this pattern.

Predicting the Next Terms

Using the constant difference, we can find the next terms in the difference sequences and finally the next term in the original series:

1. The next third difference will be \(24 + 6 = 30\).
2. The next second difference will be the last second difference plus the next third difference: \(60 + 30 = 90\).
3. The next first difference will be the last first difference plus the next second difference: \(120 + 90 = 210\).
4. The next term in the original series will be the last term plus the next first difference: \(215 + 210 = 425\).

Alternative Pattern for First Differences

The first differences (0, 6, 24, 60, 120) can also be represented by the formula \(n³ - n\), where \(n\) corresponds to the position in the difference sequence starting from 1:

• For \(n=1\): \(1³ - 1 = 1 - 1 = 0\)
• For \(n=2\): \(2³ - 2 = 8 - 2 = 6\)
• For \(n=3\): \(3³ - 3 = 27 - 3 = 24\)
• For \(n=4\): \(4³ - 4 = 64 - 4 = 60\)
• For \(n=5\): \(5³ - 5 = 125 - 5 = 120\)

The next first difference (for \(n=6\)) would be \(6³ - 6 = 216 - 6 = 210\). This is the difference between the 7th and 6th terms. So, the 7th term is \(215 + 210 = 425\).

Conclusion

Both methods of analyzing the differences lead to the same result. The number that replaces the question mark is 425.

Revision Table: Detailed Series Analysis

Additional Information: Solving Number Series Problems

Number series questions are common in logical reasoning and quantitative aptitude tests. To solve them effectively, consider these tips:

• Always calculate the differences between consecutive terms. If the first differences don't show a clear pattern, calculate the differences of the differences (second differences), and so on.
• Look for patterns involving arithmetic progression, geometric progression, squares, cubes, or combinations.
• Sometimes, the pattern might involve alternating operations (addition and subtraction, multiplication and division) or two interleaved series.
• Practice is key to quickly recognizing different types of series patterns.

This particular problem is an example where analyzing differences multiple times reveals a constant difference at the third level, indicating a polynomial pattern.