Problem 11 - Calculus for the Life Sciences: A Modeling Approach Volume I

Question

Add the equations, $$\begin{aligned} H_{2}-H_{1} &=\frac{-849}{2}\left(t_{2}^{2}-t_{1}^{2}\right)+126\left(t_{2}-t_{1}\right) \\\ H_{3}-H_{2} &=\frac{-849}{2}\left(t_{3}^{2}-t_{2}^{2}\right)+126\left(t_{3}-t_{2}\right) \end{aligned}$$ $$H_{n}-H_{n-1}=\frac{-849}{2}\left(t_{n}^{2}-t_{n-1}^{2}\right)+126\left(t_{n}-t_{n-1}\right) $$to obtain $$H_{n}-H_{1}=\frac{-849}{2}\left(t_{n}^{2}-t_{1}^{2}\right)+126\left(t_{n}-t_{1}\right)$$

Step-by-Step Solution

Answer

Add the equations using their telescoping nature to get: \(H_n - H_1 = \frac{-849}{2} (t_n^2 - t_1^2) + 126 (t_n - t_1)\).

Step 1: Understand the Problem Statement

We are given a series of equations for consecutive terms starting from \( H_2 - H_1 \) to \( H_n - H_{n-1} \). Our task is to add these equations together to find the general relation between \( H_n \) and \( H_1 \).

Step 2: Set Up the Summation for H Differences

Identify that we need to add the series of equations to find the total difference \( H_n - H_1 \). Note the pattern in each equation for differences \( H_k - H_{k-1} \). The left-hand side of the sum will simply be \( \sum (H_k - H_{k-1}) = H_n - H_1 \).

Step 3: Sum the Right-Hand Side of the Equations

Incorporate the computations for both terms in the given equations. The sums of the right-hand sides are \( \sum \left( \frac{-849}{2} (t_k^2 - t_{k-1}^2) + 126 (t_k - t_{k-1}) \right) \).

Step 4: Simplify Using Telescoping Nature

Recognize that this is a telescoping series. When you add up like this, most terms cancel out, leaving just the first and the last terms: \( \frac{-849}{2} (t_n^2 - t_1^2) + 126 (t_n - t_1) \).

Step 5: Write the Final Summation Equation

Combine the results from the telescoping nature of the series. The ultimate expression is \( H_n - H_1 = \frac{-849}{2} (t_n^2 - t_1^2) + 126 (t_n - t_1) \).

Key Concepts

Summation Techniques

Summation techniques are pivotal when dealing with series of equations or terms that need to be combined. In this exercise, we work with a series of equations that express the difference between consecutive terms:

• \( H_2 - H_1 \)
• \( H_3 - H_2 \)
• and so on, up to \( H_n - H_{n-1} \).

To arrive at the sum \( H_n - H_1 \), we add these equations one onto the next. The key to solving this efficiently lies in recognizing repeating patterns and simplifying them through addition.This is where our summation technique comes into play. It involves identifying the change from one term to the next as the equations are summed, allowing us to recognize that the left-hand side simplifies to a straightforward expression:

• \( \sum (H_k - H_{k-1}) = H_n - H_1 \) simply collapses once all equations are summed.

With this, the left-hand side of the summation becomes fully evident, setting the stage to focus on resolving the second parts, related to both quadratic and linear sequences.

Difference Equations

Difference equations are equations that express the difference between two successive terms. In our context, they are given for each consecutive pair:

• \( H_{k} - H_{k-1} \), where \( k \) runs up to \( n \).

Each equation in our series has the structure:

• \( H_k - H_{k-1} = \frac{-849}{2}(t_k^2 - t_{k-1}^2) + 126(t_k - t_{k-1}) \).

By summing these difference equations, we aim to determine a general relationship between the very first and the \( n^{th} \) term in the series.Each difference equation provides a snapshot of how each term in the series is derived from the previous one. This cumulative understanding allows us to unravel the overall difference between the starting term and the desired final term, \( H_n \) from \( H_1 \).

Mathematical Simplification

Mathematical simplification is a crucial part of solving telescoping series and involves eliminating repetitive terms. In a telescoping series, like the one presented here, terms "collapse" as adjacent terms cancel each other out. This is especially evident on the right-hand side of our equations:
- When we sum the terms \( \frac{-849}{2}(t_k^2 - t_{k-1}^2) + 126(t_k - t_{k-1}) \), intermediate terms vanish.Essentially,

• the internal terms of the sequence nullify, leaving us with only the difference between the very first and the last terms.
• This means our final equation becomes \( \frac{-849}{2}(t_n^2 - t_1^2) + 126(t_n - t_1) \) as the only survivors.

The act of simplification reduces complex expressions into more manageable forms, unveiling the neat, simple desired relation \( H_n - H_1 \), which uncovers the entire transformational nature of the equation.