Finding the Fifth Number in a Sequence: A Mathematical Puzzle

In this article, we explore a mathematical problem involving nine numbers. We delve into the methods and reasoning behind calculating a specific number in the sequence given the average values of certain segments. The puzzle is not only engaging but also provides a clear understanding of how averages and arithmetic sequences work.

Mathematical Problem Statement

We are given nine numbers, represented as a1 a2 a3 a4 a5 a6 a7 a8 a9. Three key conditions are provided:

The average of the first five numbers is 20. The average of the last five numbers is also 20. The average of all nine numbers is 16.

The problem requires us to find the fifth number, a5.

Step-by-Step Solution

Step 1: Calculating the Total for the First Five Numbers

The average of the first five numbers is 20. Therefore:

Middle Math Note: calculating the total

Math: sum(a1, a2, a3, a4, a5) 20 * 5 100

Step 2: Calculating the Total for the Last Five Numbers

The average of the last five numbers is also 20. Therefore:

Middle Math Note: calculating the total

Math: sum(a5, a6, a7, a8, a9) 20 * 5 100

Step 3: Calculating the Total for All Nine Numbers

The average of all nine numbers is 16. Therefore:

Middle Math Note: calculating the total

Math: sum(a1, a2, a3, a4, a5, a6, a7, a8, a9) 16 * 9 144

Step 4: Combining and Analyzing the Totals

Let’s add the totals of the first five and the last five numbers:

Math: sum(a1 a2 a3 a4 a5) sum(a5 a6 a7 a8 a9) 100 100 200

Since the sum of all nine numbers is 144, we can set up the following equation:

Middle Math Note: solving for the fifth number

Math: 200 - sum(a1 a2 a3 a4 a6 a7 a8 a9) 144

Therefore:

Math: sum(a1 a2 a3 a4 2a5 a6 a7 a8 a9) 200 - 144 56

Simplifying:

Math: 2a5 56 - 144 56

Therefore:

Math: a5 56 / 2 28

Conclusion

The fifth number in the sequence is 28. This result is derived from the given conditions and follows a systematic approach to solving similar problems involving averages and arithmetic sequences.

Understanding the above steps helps in breaking down complex problems into simpler, manageable parts. This method can be applied to various similar problems in mathematics and is a valuable skill for students and professionals in fields requiring analytical thinking and problem-solving.