Understanding the Linear Equation: aₙ = 3 + (n – 1) · 6 = 6n – 3

Linear equations are fundamental building blocks in mathematics, appearing in algebra, calculus, statistics, and real-world modeling. One such equation—aₙ = 3 + (n – 1) · 6—is a classic example of an arithmetic sequence in standard form. In this SEO-optimized article, we’ll break down how to interpret this formula, convert it into its common representation, and explore its practical applications. Whether you're a student, educator, or self-learner, understanding this equation enhances your grasp of sequences, linear progressions, and their role in mathematical modeling.

Understanding the Context

What Is the Equation aₙ = 3 + (n – 1) · 6?

The expression aₙ = 3 + (n – 1) · 6 defines a linear recurrence relation commonly used to model arithmetic sequences—sequences where each term increases by a constant difference. Here’s what each component means:

aₙ: Represents the nth term in the sequence
n: The position or index (starting at 1)
3: The first term (when n = 1)
6: The common difference between consecutive terms
(n – 1) · 6: Accounts for progression—each step adds 6

Example:
For n = 1:
a₁ = 3 + (1 – 1) · 6 = 3 + 0 = 3

$If 1 \cdot 3 + 2 \cdot 3^2 + 3 \cdot 3^3 + \cdots + n \cdot 3^n = (2n-1)$

Image Gallery

$If 1 \cdot 3 + 2 \cdot 3^2 + 3 \cdot 3^3 + \cdots + n \cdot 3^n = (2n-1)$

Solved (a) 3⋅2⋅31+3⋅3⋅41+3⋅4⋅51+⋯+3n(n+1)1=6(n+1)n−1, for | Chegg.com

Exercise 6.7.6. (a) Let cn=2⋅4⋅6⋯2n1⋅3⋅5⋯(2n−1) for | Chegg.com

Solved The solutions to an=−3an−1+18an−2 have the form | Chegg.com

The sequence a1=6,an=3an-1 can also be written | Chegg.com

Key Insights

For n = 2:
a₂ = 3 + (2 – 1) · 6 = 3 + 6 = 9

For n = 3:
a₃ = 3 + (3 – 1) · 6 = 3 + 12 = 15

So the sequence begins: 3, 9, 15, 21, 27,... where each term increases by 6.

Converting to Standard Form: aₙ = 6n – 3

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Final Thoughts

To simplify analysis, we convert the recurrence into standard form for arithmetic sequences:
aₙ = a₁ + (n – 1)d, where:

a₁ = 3 (first term)
d = 6 (common difference)

Substituting:
aₙ = 3 + (n – 1) · 6
= 3 + 6n – 6
= 6n – 3

This linear function models aₙ as a direct variable of n, making it easy to compute any term without recursion. For instance:

To find the 10th term: a₁₀ = 6×10 – 3 = 57
The relationship is linear with slope 6 and y-intercept –3, visually represented on a graph.

The Mathematics Behind the Formula

The general structure aₙ = A + (n – 1)d is derived from:

Starting at A = 3 (the base value)
Building the sequence by repeatedly adding d = 6
The closed-form formula avoids recalculating prior terms, offering O(1) time complexity for term lookup.

This form is essential in:

Financial modeling (e.g., linear profit growth)
Physics (constant velocity motion)
Computer science (iteration counts)
Statistics (linear regression slopes)

Applications of aₙ = 6n – 3

Real-World Use Cases

Budget Projections: If monthly expenses increase by a fixed amount, this equation models total spend over time.
Distance Travel: A vehicle moving at constant speed covers distance d = vt; here, aₙ could represent total distance after n hours with initial offset.
Salary Growth: Stepwise raises based on fixed increments per year.