Arithmetic Sequences: Finding the Fourth Term Given the First Three Terms

Understanding arithmetic sequences is a fundamental concept in mathematics that can be applied in various fields such as finance, physics, and even everyday problem-solving. This article will delve into finding the fourth term of an arithmetic sequence given the first three terms. We will explore the steps involved in solving for the common difference and the variable x. Let's proceed step-by-step.

The First Three Terms of an Arithmetic Sequence

Given the first three terms of an arithmetic sequence as:

a1 x-3 a2 x2/259 - 9 a3 3x-11

To find the fourth term, a4, we first need to determine the common difference, d, and the value of x.

Step-by-Step Solution

Step 1: Set up the equations for the common difference.

The common difference can be expressed as:

d a2 - a1 a3 - a2

Let's set up the equations:

d x2/25 - 9 - (x - 3) x2/25 - x - 6 d 3x - 11 - (x2/25 - 9) 3x - x2/25 - 2

Step 2: Set the two expressions for d equal to each other.

From the previous expressions:

x2/25 - x - 6 3x - x2/25 - 2

Multiplying the entire equation by 25 to eliminate the fraction:

x2 - 25x - 150 75x - 50 - x2

Combining like terms:

2x2 - 10 - 100 0

Dividing the entire equation by 2:

x2 - 5 - 50 0

Step 3: Solve for x.

Using the quadratic formula, x (-b ± sqrt(b2 - 4ac)) / (2a):

a 1 b -50 c -50

The discriminant:

b2 - 4ac (-50)2 - 4 * 1 * (-50) 2500 200 2700

Substituting back into the quadratic formula:

x (50 ± sqrt(2700)) / 2

Calculating the two possible values:

x (50 sqrt(2700)) / 2 40 x (50 - sqrt(2700)) / 2 10

Step 4: Calculate the fourth term for both values of x.

Case 1: x 40 a1 40 - 3 37 a2 402/25 - 9 1600/25 - 9 64 - 9 73 a3 3 * 40 - 11 120 - 11 109 The common difference d: 73 - 37 36 The fourth term a4: 109 36 145 Case 2: x 10 a1 10 - 3 7 a2 102/25 - 9 100/25 - 9 4 - 9 13 a3 3 * 10 - 11 30 - 11 19 The common difference d: 13 - 7 6 The fourth term a4: 19 6 25

Final Answers:

Thus, the fourth term can be either 145 when x 40 or 25 when x 10.

Conclusion

Arithmetic sequences can seem complex, but breaking down the problem into steps and using algebraic techniques helps in solving them. In this article, we have learned how to find the fourth term of an arithmetic sequence given the first three terms. We also explored how to solve for the variable x and the common difference d. Understanding these steps is crucial for problem-solving in various mathematical contexts.