Problem 12 - Calculus for Biology and Medicine

Question

Determine the values of the sequence \(\left|a_{n}\right|\) for \(n=0,1,2, \ldots, 5\) $$ a_{n}=n^{3} \sqrt{n+1} $$

Step-by-Step Solution

Answer

|a_0|=0, |a_1|=\sqrt{2}, |a_2| \approx 13.856, |a_3|=54, |a_4| \approx 143.108, |a_5| \approx 306.186.

Step 1: Understanding the Expression for the Sequence

The given sequence is \(a_n = n^3 \sqrt{n+1}\). We need to find the values of \(|a_n|\), which are the absolute values of each \(a_n\) for \(n = 0, 1, 2, 3, 4, 5\). Since \(a_n\) is non-negative for all \(n \geq 0\), \(|a_n| = a_n\). Thus, we simply need to compute \(n^3 \sqrt{n+1}\) for each specified \(n\).

Step 2: Compute for \(n=0\)

Substitute \(n = 0\) into the expression: \[ a_0 = 0^3 \sqrt{0+1} = 0 \cdot 1 = 0 \] Therefore, \(|a_0| = 0\).

Step 3: Compute for \(n=1\)

Substitute \(n = 1\) into the expression: \[ a_1 = 1^3 \sqrt{1+1} = 1 \cdot \sqrt{2} = \sqrt{2} \] Therefore, \(|a_1| = \sqrt{2}\).

Step 4: Compute for \(n=2\)

Substitute \(n = 2\) into the expression: \[ a_2 = 2^3 \sqrt{2+1} = 8 \cdot \sqrt{3} \approx 13.856 \] Therefore, \(|a_2| \approx 13.856\).

Step 5: Compute for \(n=3\)

Substitute \(n = 3\) into the expression: \[ a_3 = 3^3 \sqrt{3+1} = 27 \cdot 2 = 54 \] Therefore, \(|a_3| = 54\).

Step 6: Compute for \(n=4\)

Substitute \(n = 4\) into the expression: \[ a_4 = 4^3 \sqrt{4+1} = 64 \cdot \sqrt{5} \approx 143.108 \] Therefore, \(|a_4| \approx 143.108\).

Step 7: Compute for \(n=5\)

Substitute \(n = 5\) into the expression: \[ a_5 = 5^3 \sqrt{5+1} = 125 \cdot \sqrt{6} \approx 306.186 \] Therefore, \(|a_5| \approx 306.186\).

Key Concepts

Absolute Value

The concept of absolute value is one of the fundamental building blocks in mathematics. It represents the distance of a number on the number line from zero, without considering its direction. This means that absolute value is always non-negative. For any real number \( x \), the absolute value is denoted as \(|x|\). Here are a few core points to grasp:

• If \( x \) is positive, \(|x| = x\).
• If \( x \) is negative, \(|x| = -x\).
• If \( x \) is zero, \(|x| = 0\).

In our exercise with the sequence \(a_n = n^3 \sqrt{n+1}\), we are tasked with finding \(|a_n|\). Since \(a_n\) is constructed using non-negative terms for all \(n \geq 0\), the absolute value \(|a_n|\) simplifies to \(a_n\) itself. This is because both \(n^3\) and the square root function always yield non-negative results.
The simplification of absolute value in this context is essential, as it allows us to directly evaluate the sequence without additional considerations for positivity or negativity.

Mathematical Sequences

Mathematical sequences are ordered lists of numbers that follow a particular pattern or rule. A sequence is typically denoted as \((a_n)\), where \(n\) represents a non-negative integer which is often called the "index."

• The formula \(a_n = n^3 \sqrt{n+1}\) shows how each term in the sequence is defined.
• The sequence begins with \(n=0\) and extends onward as per the given range.

Sequences can express complex mathematical problems, while still being rooted in fairly straightforward rules.
Each term is determined by increasing \(n\) incrementally. The unique aspect of this sequence is its dependence on both a polynomial expression \(n^3\) and a radical function \(\sqrt{n+1}\). This mixture results in rapid growth, as seen in increasing values such as \(|a_3| = 54\) and \(|a_5| \approx 306.186\).
Understanding how sequences grow and behave is crucial, as it enhances our insight into the behavior of functions and helps us anticipate the characteristics of more complex mathematical models.

Step-by-Step Computation

Solving complex problems in mathematics involves breaking them down into smaller, more manageable steps. This step-by-step approach is beneficial, especially in sequential problems where each term builds on the prior calculations.
Here's how a step-by-step computation is helpful:

• It simplifies calculations, making them less error-prone.
• It enhances understanding by sequentially approaching each component of the problem.

In our exercise with \(a_n = n^3 \sqrt{n+1}\), each value of \(n\) from 0 to 5 was substituted individually, making the computation clearer and more organized. Each step elaborated on:

• Substitution: Replacing \(n\) in \(n^3 \sqrt{n+1}\) with the respective integer values.
• Calculation: Performing arithmetic or simplifications needed to finalize the expression.
• Result: Concluding each step with its specific \(|a_n|\) value, ensuring clarity and correctness.

Such methodical approaches not only solve the problem but also instill confidence in learners, providing them with a replicable strategy for tackling similar problems in the future.