Identifying the Missing Number in a Unique Arithmetic Sequence: A Comprehensive Guide

When faced with a challenging number series, it is essential to understand the underlying pattern to accurately determine the missing number. In this article, we will explore how to identify the missing number in a specific number series and present the solution step-by-step.

Given Series:

Let's take a closer look at the given number series: 611213656. The task is to find the missing number that comes after 65.

Understanding the Pattern

The sequence follows a unique and consistent pattern. Each term in the series is generated by multiplying a specific number and then adding the result of multiplying that number by a sequence of increasing integers.

Pattern Analysis

Let's break down the series to identify the pattern:

61 × 1 61 1 62 - 1 61 (11)

115 × 2 1110 2 1121 - 10 21 (115 × 2 - 10)

215 × 3 2115 3 2118 - 15 36 (215 × 3 - 15)

365 × 4 3620 4 3624 - 20 56 (365 × 4 - 20)

565 × 5 5625 5 5630 - 25 81 (565 × 5 - 25)

The general pattern can be described as follows:

term_n term_p × (n 1) - n * 5

Breaking Down the Steps

Let's break down the steps to better understand the solution:

Identify the initial terms and their operations.

Determine the incremental changes in the multipliers and additions.

Apply the identified pattern to find the missing number.

Step-by-Step Solution

1. Start with the first term: 611213656.
2. Identify the pattern of the operations:

61 × 1 61 1 62 - 1 61 (11)

115 × 2 1110 2 1121 - 10 21 (115 × 2 - 10)

215 × 3 2115 3 2118 - 15 36 (215 × 3 - 15)

365 × 4 3620 4 3624 - 20 56 (365 × 4 - 20)

565 × 5 5625 5 5630 - 25 81 (565 × 5 - 25)

3. Apply the identified pattern to find the next term:

565 × 5 5625 5 5630 - 25 81 (565 × 5 - 25)

Explanation and General Formula

From the given series, we observe that the difference between consecutive terms follows a pattern:

5, 10, 15, 20, 25

This implies that the next term will be:

56 25 81

Conclusion

To find the missing number in the series, we need to recognize the arithmetic pattern and apply it correctly. In this case, the missing number is 81, making the complete series 61121365681.

Additional Series for Practice

Let's explore another series to practice identifying patterns:

Series 1: 69121518
Series 2: 65111713

These series can be solved by following the same pattern recognition and step-by-step analysis method.

For Series 1: The pattern is simply an addition of the next odd number (3, 7, 11, 15, 19).

For Series 2: The pattern is to add the next odd number to the previous number (15, 17, 19, 21, 23).

Key Takeaways

Recognize the unique arithmetic pattern in the series.

Apply the pattern consistently to find the missing number.

Break down the series into components for clearer understanding.

Final Answer

The missing number in the sequence 611213656 is 81.