Discover the Next Number in a Series: Unraveling the Mathematical Mystery

Mathematics is filled with intriguing patterns and sequences that can challenge our understanding. One interesting sequence is the series: 5, 13, 25, 41, 61, 85, 113, 145, _. Let's explore how we can determine the next number in the series while gaining insights into patterns and mathematical functions.

Understanding the Series

The sequence in question is: 5, 13, 25, 41, 61, 85, 113, 145, _. To solve this, we'll employ a step-by-step approach, examining the differences and identifying any underlying patterns or functions.

Examine Differences

The first step is to calculate the differences between consecutive terms in the series:

13 - 5 8 25 - 13 12 41 - 25 16 61 - 41 20 85 - 61 24 113 - 85 28 145 - 113 32

From this, we have the sequence of differences: 8, 12, 16, 20, 24, 28, 32.

Identify the Pattern in Differences

Next, we look at the differences of these differences:

12 - 8 4 16 - 12 4 20 - 16 4 24 - 20 4 28 - 24 4 32 - 28 4

The second differences are constant at 4, indicating a quadratic function is at play. This pattern is a clear sign that the sequence is generated by a quadratic equation.

Determine the Next Term

To find the next difference, we add 4 to the last difference:

32 4 36

Then, we add this difference to the last term in the series:

145 36 181

Thus, the next number in the series is 181.

Pattern Recognition in Series

Understanding patterns in number series like these can provide valuable insights into mathematical functions. In this series, we observed that:

The differences between consecutive terms form a sequence that increases by 4 each step. The differences are multiples of 4.

By recognizing such patterns, we can predict the next term and often the function behind the sequence.

Step-by-Step Breakdown

Given series: 5, 13, 25, 41, 61, 85, 113, 145, …

Step 1: Calculate the differences between consecutive terms:

13 - 5 8 25 - 13 12 41 - 25 16 61 - 41 20 85 - 61 24 113 - 85 28 145 - 113 32

Step 2: Calculate the differences of these differences:

12 - 8 4 16 - 12 4 20 - 16 4 24 - 20 4 28 - 24 4 32 - 28 4

Step 3: Add 4 to the last difference to find the next difference:

32 4 36

Step 4: Add this difference to the last term in the series to find the next term:

145 36 181

Thus, the next term is 181.

By breaking down the problem, we can identify patterns and use them to predict the next term in a series, a skill valuable in mathematics and beyond.