Solving for the First Term of an Arithmetic Sequence

Arithmetic sequences are a fundamental part of mathematics, often appearing in various forms of problem-solving. This article explores a specific type of problem where we are given the seventh term and the common difference, and our goal is to determine the first term. Understanding these concepts not only aids in solving such problems but also enhances one's skills in manipulating algebraic expressions and understanding the nature of sequences.

Understanding the Problem

Consider the problem where the seventh term of an arithmetic sequence is 84, and the common difference is 6. To solve for the first term, we need to understand the properties of arithmetic sequences and how to apply them to find the required term.

Solving the Problem

The general formula for the n-th term of an arithmetic sequence is given by: [text{an} text{a1} (text{n} - 1)d]

Where:

an is the n-th term, a1 is the first term, d is the common difference, n is the term number.

Substituting the Given Values

In this problem, we know that:

(text{a7} 84) (d 6) (text{n} 7)

Substitute these values into the formula to solve for ( text{a1} ).

Solution

First, let's substitute the formula: [text{a7} text{a1} (7 - 1) times 6] Simplifying the equation: [84 text{a1} 6 times 6] Further simplification: [84 text{a1} 36] Isolating (text{a1}): [text{a1} 84 - 36] Therefore, (text{a1} 48).

Additional Insights

The first term of the arithmetic sequence is 48. The sequence, given the common difference of 6, can be written as follows:

[48, 54, 60, 66, 72, 78, 84, 90, 96, 102, ldots]

General Term of an Arithmetic Sequence

The general term ( text{tn} ) of an arithmetic sequence can be expressed as:

[ text{tn} text{a1} (n - 1)d ]

Substituting the known values, we get:

[ text{tn} 48 (n - 1) times 6 ]

Simplifying further, we can express it as:

[ text{tn} 6n 42 ]

Conclusion

By understanding the basic formula for the n-th term of an arithmetic sequence, we can solve for the first term given any term and the common difference. This process not only enhances problem-solving skills but also deepens the understanding of mathematical sequences and their applications.