Finding the Missing Number in the Series

The question asks us to find the missing number in the given series: 16, 17, 25, 52, 116, ?

To solve number series questions, we usually look for a pattern in the differences between consecutive terms, the ratios between terms, or a specific mathematical operation applied sequentially.

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

• Difference between the 2nd and 1st term: \(17 - 16 = 1\)
• Difference between the 3rd and 2nd term: \(25 - 17 = 8\)
• Difference between the 4th and 3rd term: \(52 - 25 = 27\)
• Difference between the 5th and 4th term: \(116 - 52 = 64\)

The sequence of differences is 1, 8, 27, 64. Let's analyze this new sequence to find a pattern.

We can observe that these differences are perfect cubes:

• \(1 = 1 \times 1 \times 1 = 1^3\)
• \(8 = 2 \times 2 \times 2 = 2^3\)
• \(27 = 3 \times 3 \times 3 = 3^3\)
• \(64 = 4 \times 4 \times 4 = 4^3\)

The pattern in the differences is the cube of consecutive natural numbers, starting from 1. The next difference in the series should be the cube of the next natural number, which is 5.

The next difference will be \(5^3\).

\(5^3 = 5 \times 5 \times 5 = 125\)

To find the missing number in the original series, we need to add this next difference (125) to the last term of the original series (116).

Missing number = Last term + Next difference

Missing number = \(116 + 125\)

Let's perform the addition:

So, the missing number in the series is 241.

Step-by-Step Solution to Find the Missing Number

1. Write down the given series: 16, 17, 25, 52, 116, ?.
2. Calculate the difference between each consecutive pair of numbers.
3. Differences: \(17-16=1\), \(25-17=8\), \(52-25=27\), \(116-52=64\).
4. Examine the series of differences: 1, 8, 27, 64.
5. Identify the pattern in the differences: \(1=1^3\), \(8=2^3\), \(27=3^3\), \(64=4^3\). The pattern is the cube of consecutive integers.
6. Determine the next term in the difference series. Following the pattern, the next difference should be \(5^3\).
7. Calculate the next difference: \(5^3 = 125\).
8. Add the next difference to the last term of the original series to find the missing number: \(116 + 125\).
9. Calculate the sum: \(116 + 125 = 241\).
10. The missing number is 241.

Summary of the Series Pattern

Adding the next difference (125) to the 5th term (116) gives \(116 + 125 = 241\).

Therefore, the missing number is 241.

Revision Table: Number Series Patterns

Understanding common number series patterns is key to solving such questions.

Additional Information: Solving Number Series Questions

Here are some tips for approaching number series problems:

• Look for Simple Patterns First: Check for arithmetic or geometric progressions as they are the easiest to spot.
• Calculate Differences: If a simple pattern isn't obvious, find the differences between consecutive terms. This often reveals a new series with a simpler pattern (like the squares or cubes of numbers as seen in this problem).
• Calculate Ratios: For increasing series, check ratios if differences don't follow a clear pattern. This helps identify geometric progressions or related patterns.
• Look for Squares or Cubes: Be familiar with perfect squares and cubes; they frequently appear in series patterns, sometimes with slight modifications (n²+1, n³-1, etc.).
• Consider Alternating Patterns: If the series alternates between increasing and decreasing, there might be two interleaved series.
• Check for Common Mathematical Operations: Sometimes the pattern involves multiplication, division, addition, or subtraction that changes in a predictable way, or a combination of operations.
• Practice: Solving many different types of number series problems helps you quickly recognize common patterns.