Introduction

Understanding and solving problems related to averages and sequences is a fundamental skill in mathematics. One common problem involves finding a specific number in a sequence based on given averages. Let's explore a challenging problem and provide a detailed solution to help improve your comprehension and problem-solving skills.

Problem Statement

Given seven numbers (a_1, a_2, a_3, a_4, a_5, a_6, a_7), the average of the first four numbers is 4, and the average of the last four numbers is also 4. If the average of these seven numbers is 3, then what is the fourth number (a_4)?

Understanding the Problem

We need to find the value of (a_4) given the constraints on the averages of the numbers. Let's break down the problem step-by-step.

Step 1: Setting Up Equations for Averages

The average of the first four numbers is given by:

[frac{a_1 a_2 a_3 a_4}{4} 4]

Thus, we can write:

[a_1 a_2 a_3 a_4 16 quad text{(Equation 1)}]

The average of the last four numbers is given by:

[frac{a_4 a_5 a_6 a_7}{4} 4]

Thus, we can write:

[a_4 a_5 a_6 a_7 16 quad text{(Equation 2)}]

The average of all seven numbers is given by:

[frac{a_1 a_2 a_3 a_4 a_5 a_6 a_7}{7} 3]

Thus, we can write:

[a_1 a_2 a_3 a_4 a_5 a_6 a_7 21 quad text{(Equation 3)}]

Step 2: Using Equations to Find (a_4)

We now use the equations to find (a_4).

From Equation 1:

[a_1 a_2 a_3 a_4 16]

From Equation 2, we can express (a_5 a_6 a_7):

[a_5 a_6 a_7 16 - a_4 quad text{(Equation 4)}]

Substituting Equation 4 into Equation 3:

[16 (16 - a_4) 21]

Simplifying this:

[32 - a_4 21]

Thus,

[a_4 32 - 21 11]

Therefore, the fourth number (a_4) is 11.

Alternative Methods

There are alternative methods to solve this problem, and exploring different approaches can enhance your understanding and problem-solving skills.

Method 1: Direct Subtraction

Average of 7 numbers 3, so total 21.

Average of first 4 numbers 5, so total 20.

Total of last 3 numbers 28 - 20 8.

Average of last 4 numbers 5, so total 20.

Total of last 3 numbers 20 - 8 12.

Thus, the fourth number 20 - 8 12.

Method 2: Summing Totals

Average of 7 numbers 3, so total 21.

Total of first 4 numbers 20.

Total of last 4 numbers 20.

Sum of first 4 and last 4 numbers 40.

Fourth number 40 - 21 12.

Explanation of Steps Involved

In these methods, the key is to use the properties of averages and the given constraints to simplify the problem.

Conclusion

To summarize, the fourth number in the sequence is 11. Understanding how to use averages and constraints to solve problems can greatly enhance your problem-solving skills in mathematics. Whether you use equations or direct subtraction, the key is to break down the problem step-by-step and substitute values carefully.

Keywords

keywords: average, sequence, arithmetic