Finding the Missing Number in a Series

This question asks us to find the missing number in the given series: 2, 3, 7, 16, 32, ?

To solve number series problems, we look for a pattern or a rule that connects the terms. This pattern could involve addition, subtraction, multiplication, division, squares, cubes, or a combination of operations.

Analyzing the Series Pattern

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

• Difference between the 2nd and 1st term: ${3 - 2 = 1}$
• Difference between the 3rd and 2nd term: ${7 - 3 = 4}$
• Difference between the 4th and 3rd term: ${16 - 7 = 9}$
• Difference between the 5th and 4th term: ${32 - 16 = 16}$

The sequence of differences we found is: 1, 4, 9, 16.

Identifying the Difference Pattern

Let's look closely at the sequence of differences: 1, 4, 9, 16. We can see that these numbers are perfect squares:

• ${1 = 1^2}$
• ${4 = 2^2}$
• ${9 = 3^2}$
• ${16 = 4^2}$

The pattern in the differences is that the difference between the n-th term and the (n-1)-th term is the square of (n-1). For example, the difference between the 5th term and the 4th term is $16$, which is $4^2$, where $n=5$ and $n-1=4$.

Predicting the Next Term

Following this pattern, the next difference (between the 6th term and the 5th term) should be the next perfect square, which is $5^2$.

• The next difference should be ${5^2 = 25}$.

To find the missing number (the 6th term), we add this difference to the last known term (the 5th term), which is 32.

• Missing number = 5th term + next difference
• Missing number = ${32 + 25}$
• Missing number = ${57}$

Verification of the Series Pattern

Let's write out the series using the identified pattern:

• Term 1: 2
• Term 2: ${2 + 1^2 = 2 + 1 = 3}$
• Term 3: ${3 + 2^2 = 3 + 4 = 7}$
• Term 4: ${7 + 3^2 = 7 + 9 = 16}$
• Term 5: ${16 + 4^2 = 16 + 16 = 32}$
• Term 6: ${32 + 5^2 = 32 + 25 = 57}$

The calculated sequence matches the given series up to the last known term, and the next term fits the pattern.

Conclusion

Based on the pattern where the difference between consecutive terms is the square of the term number (starting from $1^2$ for the difference between term 2 and term 1), the missing number in the series 2, 3, 7, 16, 32, ? is 57.

Number Series Problem Revision

When tackling number series questions, always start by finding the difference between consecutive terms. If that doesn't reveal a pattern immediately, look for patterns in the differences themselves, or consider other operations like multiplication, division, or combinations. Sometimes, the pattern might involve alternating operations or multiple interacting series.

Additional Information on Logical Reasoning

Number series problems are common in logical reasoning and quantitative aptitude tests. They assess your ability to identify patterns and apply logical rules. Practicing various types of series (arithmetic, geometric, Fibonacci, differences, squares, cubes, etc.) helps improve problem-solving speed and accuracy.