Finding Common Difference And First Term In An Arithmetic Progression

Arithmetic Progressions (APs) are fundamental sequences in mathematics where the difference between consecutive terms remains constant. This constant difference is known as the common difference. Understanding APs is crucial for various mathematical applications and problem-solving scenarios. In this article, we will delve into a specific problem involving an AP and systematically determine its common difference and first term.

Understanding Arithmetic Progressions

Before diving into the problem, let's solidify our understanding of arithmetic progressions. An AP is characterized by a sequence of numbers where each term is obtained by adding a fixed number (the common difference) to the previous term. The general form of an AP can be represented as:

• a, a + d, a + 2d, a + 3d, ...

Where:

• 'a' is the first term of the sequence.
• 'd' is the common difference.

Each term in the AP can be expressed using the formula:

• an = a + (n - 1)d

Where:

• an represents the nth term of the AP.

This formula is essential for solving problems related to arithmetic progressions, as it allows us to relate any term in the sequence to the first term and the common difference. With a clear grasp of these concepts, let's tackle the problem at hand.

Problem Statement: Unraveling the Sequence

The problem states: In an AP, the difference between the 8th and the 4th term is 20, and the 8th term is 1½ times the 4th term. Our objective is to determine:

(i) The common difference (ii) The first term of the sequence

This problem presents us with two crucial pieces of information that we can use to form equations. By translating these statements into mathematical expressions, we can systematically solve for the unknowns – the common difference and the first term. Let's break down each piece of information and formulate the corresponding equations.

Translating the Problem into Equations

The key to solving this problem lies in translating the given information into mathematical equations. We have two statements:

1. "The difference between the 8th and the 4th term is 20"
   • Using the formula for the nth term, we can express the 8th term as a + 7d and the 4th term as a + 3d. The difference between them is:
     • (a + 7d) - (a + 3d) = 20
2. "The 8th term is 1½ times the 4th term"
   • Converting 1½ to a fraction, we get 3/2. This statement translates to:
     • a + 7d = (3/2)(a + 3d)

Now we have a system of two equations with two unknowns, 'a' and 'd'. Solving this system will give us the values of the common difference and the first term.

Simplifying the Equations

Before we proceed to solve the system of equations, let's simplify them to make the calculations easier.

Equation 1: (a + 7d) - (a + 3d) = 20

• Simplifying, we get:
  • 4d = 20

Equation 2: a + 7d = (3/2)(a + 3d)

• Multiplying both sides by 2 to eliminate the fraction:
  • 2(a + 7d) = 3(a + 3d)
• Expanding:
  • 2a + 14d = 3a + 9d
• Rearranging:
  • a - 5d = 0

Now our system of equations is:

1. 4d = 20
2. a - 5d = 0

These simplified equations are much easier to work with. In the next section, we will solve this system to find the values of 'a' and 'd'.

Solving the System of Equations

With the simplified equations, we can now solve for the common difference (d) and the first term (a).

Equation 1: 4d = 20

• Dividing both sides by 4:
  • d = 5

We have found the common difference! Now we can substitute this value into Equation 2 to find the first term.

Equation 2: a - 5d = 0

• Substituting d = 5:
  • a - 5(5) = 0
  • a - 25 = 0
  • a = 25

Therefore, the common difference is 5, and the first term of the sequence is 25. We have successfully determined the two unknowns using the given information and the properties of arithmetic progressions.

The Arithmetic Progression Unveiled

Having found the common difference (d = 5) and the first term (a = 25), we can now fully describe the arithmetic progression. The sequence starts with 25, and each subsequent term is obtained by adding 5 to the previous term. Thus, the AP is:

• 25, 30, 35, 40, 45, ...

We can verify our solution by checking if the given conditions are satisfied:

• The 8th term is 25 + 7(5) = 60
• The 4th term is 25 + 3(5) = 40
• The difference between the 8th and 4th term is 60 - 40 = 20 (as given)
• The 8th term (60) is 1½ times the 4th term (40) (as given)

Our solution satisfies both conditions, confirming its correctness. This exercise highlights the power of translating word problems into mathematical equations and using algebraic techniques to solve them.

Conclusion: Mastering Arithmetic Progressions

In this article, we tackled a problem involving an arithmetic progression and successfully determined its common difference and first term. We began by understanding the fundamental concepts of APs, including the general form and the formula for the nth term. Then, we translated the problem statement into a system of equations, simplified them, and solved for the unknowns. Finally, we verified our solution and presented the complete arithmetic progression.

Mastering arithmetic progressions is crucial for building a strong foundation in mathematics. These sequences appear in various contexts, from simple pattern recognition to more complex mathematical models. By understanding the properties of APs and practicing problem-solving techniques, you can enhance your mathematical skills and tackle a wide range of challenges. Remember, the key is to break down problems into manageable steps, translate information into equations, and apply the relevant formulas and concepts. With practice and perseverance, you can unlock the power of arithmetic progressions and excel in mathematics.