How To Find First Term Of Arithmetic Sequence

How to Find the First Term of an Arithmetic Sequence

Finding the first term of an arithmetic sequence might seem like a simple task, but understanding the underlying principles and various methods for solving it is crucial for success in algebra and beyond. This comprehensive guide will equip you with the knowledge and skills to confidently tackle this problem, regardless of the information provided. We'll delve into different scenarios, offering clear explanations and practical examples to solidify your understanding.

Understanding Arithmetic Sequences

Before we jump into the methods, let's establish a strong foundation. An arithmetic sequence is a series of numbers where the difference between consecutive terms remains constant. This constant difference is known as the common difference, often denoted by 'd'. The terms in the sequence are typically represented by a_n, where 'n' represents the position of the term in the sequence (e.g., a₁ is the first term, a₂ is the second term, and so on).

The general formula for the nth term of an arithmetic sequence is:

a_n = a₁ + (n-1)d

Where:

• a_n is the nth term of the sequence.
• a₁ is the first term of the sequence (what we often need to find).
• n is the position of the term in the sequence.
• d is the common difference.

This formula forms the bedrock of our methods for finding the first term. Let's explore different scenarios and how to apply this formula effectively.

Scenario 1: Given the Common Difference and Another Term

This is the most straightforward scenario. If you know the common difference (d) and the value of any other term (a_n) along with its position (n), you can easily calculate the first term (a₁). Let's illustrate with an example:

Example:

The 5th term (a₅) of an arithmetic sequence is 22, and the common difference (d) is 4. Find the first term (a₁).

Solution:

1. Identify the known values: a_n = 22, n = 5, d = 4.
2. Use the formula: a_n = a₁ + (n-1)d
3. Substitute the values: 22 = a₁ + (5-1)4
4. Simplify and solve for a₁: 22 = a₁ + 16 => a₁ = 22 - 16 = 6

Therefore, the first term (a₁) of the arithmetic sequence is 6.

Scenario 2: Given Two Terms and Their Positions

If you are given the values of two terms (a_m and a_n) and their positions (m and n), you can find both the common difference and the first term.

Example:

The 3rd term (a₃) of an arithmetic sequence is 11, and the 7th term (a₇) is 23. Find the first term (a₁).

Solution:

1. Find the common difference (d): We know that a_n - a_m = (n-m)d. Substituting our values: 23 - 11 = (7-3)d => 12 = 4d => d = 3.
2. Use the formula with either term: Let's use a₃ = 11. a_n = a₁ + (n-1)d => 11 = a₁ + (3-1)3
3. Solve for a₁: 11 = a₁ + 6 => a₁ = 11 - 6 = 5

Therefore, the first term (a₁) is 5.

Scenario 3: Given the Sum of a Certain Number of Terms and the Common Difference

This scenario requires a slightly different approach. The sum of the first 'n' terms of an arithmetic sequence is given by:

S_n = n/2 [2a₁ + (n-1)d]

If you know the sum (S_n), the number of terms (n), and the common difference (d), you can solve for a₁.

Example:

The sum of the first 10 terms (S₁₀) of an arithmetic sequence is 175, and the common difference (d) is 3. Find the first term (a₁).

Solution:

1. Use the sum formula: S_n = n/2 [2a₁ + (n-1)d]
2. Substitute the values: 175 = 10/2 [2a₁ + (10-1)3]
3. Simplify: 175 = 5 [2a₁ + 27]
4. Solve for a₁: 35 = 2a₁ + 27 => 2a₁ = 8 => a₁ = 4

Therefore, the first term (a₁) is 4.

Scenario 4: Working with Recursive Formulas

Some arithmetic sequences are defined recursively, meaning a term is defined in relation to the previous term(s). A common recursive formula is:

a_n = a_n-1 + d

Finding the first term from a recursive formula often requires working backward.

Example:

An arithmetic sequence is defined recursively as a_n = a_n-1 + 5, and a₄ = 19. Find the first term (a₁).

Solution:

1. Work backwards: Since a_n = a_n-1 + 5, we can write:
   • a₃ = a₄ - 5 = 19 - 5 = 14
   • a₂ = a₃ - 5 = 14 - 5 = 9
   • a₁ = a₂ - 5 = 9 - 5 = 4

Therefore, the first term (a₁) is 4.

Advanced Techniques and Considerations

While the scenarios above cover many common situations, let's address some more advanced considerations:

Handling Negative Common Differences:

The methods described work equally well with negative common differences. Just remember to carefully handle the signs during calculations.

Dealing with Incomplete Information:

If you lack sufficient information (e.g., only one term and no common difference), you cannot uniquely determine the first term. There will be infinitely many arithmetic sequences that could satisfy the given condition.

Applications in Real-World Problems:

Arithmetic sequences appear frequently in real-world situations, such as:

• Linear growth: The growth of a plant at a constant rate per day.
• Compound interest (simplified): If interest is added at the same amount each period (though this is a simplification).
• Depreciation: The value of an asset decreasing by a fixed amount each year.

Understanding arithmetic sequences and mastering the techniques to find the first term is essential for solving problems in various fields, from finance and engineering to physics and computer science.

Conclusion

Finding the first term of an arithmetic sequence is a fundamental skill in mathematics. By understanding the basic formula, and applying it systematically to different scenarios, you'll be able to confidently solve a wide range of problems involving arithmetic sequences. Remember to always carefully identify the given information, select the appropriate formula, and methodically solve for the unknown first term. The examples and explanations provided in this guide should serve as a solid foundation for further exploration and problem-solving success. Keep practicing, and you’ll become proficient in solving these types of problems. Remember that consistent practice is key to mastering any mathematical concept.