Understanding the Number Series Problem

The question asks us to find the missing number in the given series: 31, 35, 51, 87, 151, ?, 395. To solve this type of problem, we need to identify the pattern or rule that connects consecutive terms in the series. Number series questions are common in logical reasoning and quantitative aptitude tests and often involve patterns based on addition, subtraction, multiplication, division, squares, cubes, or a combination of these operations.

Finding the Pattern in the Series

Let's examine the differences between consecutive terms in the given number series:

• Difference between the 2nd and 1st term: $35 - 31 = 4$
• Difference between the 3rd and 2nd term: $51 - 35 = 16$
• Difference between the 4th and 3rd term: $87 - 51 = 36$
• Difference between the 5th and 4th term: $151 - 87 = 64$

The sequence of differences is 4, 16, 36, 64. Let's look for a pattern in these differences.

We can observe that these differences are perfect squares:

• $4 = 2^2$
• $16 = 4^2$
• $36 = 6^2$
• $64 = 8^2$

The pattern in the differences is the square of consecutive even numbers: $2^2, 4^2, 6^2, 8^2$.

Calculating the Missing Term in the Number Series

Following the identified pattern, the next difference in the series should be the square of the next even number after 8, which is 10. So, the next difference is $10^2 = 100$.

To find the missing term (?), we add this difference to the last known term in the sequence before the question mark (151).

Missing term = Last known term + Next difference

Missing term = $151 + 100 = 251$.

Verifying the Pattern

If the missing term is 251, the next difference in the series should follow the pattern. The sequence of differences is $2^2, 4^2, 6^2, 8^2, 10^2$. The next expected difference would be the square of the next even number, which is 12. So, the next difference should be $12^2 = 144$.

Let's check if adding this difference to our calculated missing term (251) gives the final term in the series (395).

$251 + 144 = 395$.

This matches the last number in the given series. Therefore, the pattern holds true, and the calculated missing term is correct.

The number that replaces the question mark is 251.

Conclusion

By analyzing the differences between consecutive terms, we found a pattern where the differences are squares of consecutive even numbers ($2^2, 4^2, 6^2, 8^2, 10^2, 12^2$). Applying this pattern, we determined that the missing number is 251.

Revision Table: Key Concepts in Number Series

Additional Information on Solving Number Series

Solving number series problems often requires careful observation and trial-and-error to identify the underlying pattern. Here are some tips:

• Look at the differences between consecutive terms. This is often the first step and reveals arithmetic progressions, difference series, or patterns involving squares/cubes.
• If differences don't show a clear pattern, look at the differences of the differences (double difference series).
• Check for patterns involving multiplication or division if the numbers are increasing or decreasing rapidly.
• Consider patterns involving squares, cubes, prime numbers, or Fibonacci sequences.
• Sometimes the pattern might involve alternating operations or combining two different series.
• Practice with various types of series to improve your ability to recognize common patterns quickly.

Understanding these concepts and practicing different types of number series helps in solving reasoning and quantitative aptitude questions effectively.