\[ h_n = 70 - 5(n-1) \] - United Radiology

["Understanding the Linear Sequence Formula: ( h_n = 70 - 5(n - 1) )", "Mathematics is filled with elegant formulas that describe patterns and changes over time. One such formula is the linear sequence defined by:", "[
\nh_n = 70 - 5(n - 1)
\n]", "This simple equation models a decreasing arithmetic sequence, often used in contexts like depreciation, exponential decline, or stepwise reductions over defined intervals. In this article, we’ll explore the structure, meaning, and practical applications of this formula.", "---", "### What is ( h_n = 70 - 5(n - 1) )?", "The formula represents a sequence ( h_n ), where ( n ) is the term index (starting at 1), and ( h_n ) indicates the value of the ( n )-th term.", "Rewriting the expression:
\n[
\nh_n = 70 - 5n + 5 = 75 - 5n
\n]
\nThis confirms it’s a linear function with:
\n- A constant value of 70 at ( n = 1 )
\n- A common difference (slope) of -5, meaning each subsequent term decreases by 5 units
\n- A starting point (initial term) adjusted by the formula", "---", "### Breaking Down the Formula", "- ( n ): Represents the term’s position in the sequence.
\n- 70: Initial constant value.
\n- ( -5(n - 1) ): The variable component that introduces the linear decrease. As ( n ) increases by 1, ( h_n ) drops by 5.", "Using a few examples:
\n- ( n = 1 \Rightarrow h_1 = 70 - 5(1 - 1) = 70 )
\n- ( n = 2 \Rightarrow h_2 = 70 - 5(2 - 1) = 65 )
\n- ( n = 3 \Rightarrow h_3 = 70 - 5(3 - 1) = 60 )
\n- ( n = 4 \Rightarrow h_4 = 70 - 5(4 - 1) = 55 )", "Clearly, each step declines by 5: 70, 65, 60, 55, ...", "---", "### Visualizing the Sequence", "Graphically, ( h_n ) forms a straight line with:
\n- y-intercept: ( h_1 = 70 )
\n- Slope: -5 (downward slope)
\n- Rate of change: Consistent decrease of 5 units per term", "
\n(Imagine a downward-sloping line starting at (1, 70) with each step down 5 units.)", "---", "### Practicing with Applications", "The formula ( h_n = 70 - 5(n - 1) ) is valuable for modeling real-world phenomena:", "1. Depreciation Analysis:
\n Businesses often depreciate equipment linearly. Starting value 70, cost drops 5 per year.
\n - After 3 years: ( h_3 = 60 ), reflecting a 3×5 decline.", "2. Progressive Exercise Targets:
\n In fitness routines, calories burned might decrease gradually: e.g., starting at 70, dropping 5 calories per session.", "3. Temperature Drop Patterns:
\n A cooling scenario where temperature declines linearly, such as decreasing 5 degrees hourly from 70°C.", "---", "### Key Properties at a Glance", "| Term ( n ) | Value ( h_n ) |
\n|-------------|-----------------|
\n| 1 | 70 |
\n| 2 | 65 |
\n| 3 | 60 |
\n| 4 | 55 |
\n| 5 | 50 |
\n| … | ( 70 - 5(n - 1) ) |", "---", "### How to Use This Formula", "1. Identify the pattern: Recognize linear decreases of fixed incremental value.
\n2. Determine input ( n ): Reflects the order (term number) in the sequence.
\n3. Calculate output: Plug ( n ) into formula to find ( h_n ) instantly.
\n4. Graph or tabulate: Plot values to visualize the line.", "---", "### Conclusion", "The formula ( h_n = 70 - 5(n - 1) ) captures a straightforward arithmetic decrease, ideal for modeling decay, step functions, or incremental reductions. Its simplicity makes it accessible yet powerful for students, educators, and professionals alike. Whether teaching linear functions or applying them in business or science, understanding this pattern strengthens analytical and predictive skills.", "Want to explore more sequences? Discover how quadratic or geometric models expand on linear trends.", "---", "Keywords: linear sequence formula, arithmetic sequence, ( h_n = 70 - 5(n-1) ), linear decrease model, mathematical patterns, depreciation calculation, stepwise reduction, discrete mathematics."]