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

The sequence values are: \( a_0 = 0 \), \( a_1 = \sqrt{2} \), \( a_2 = 8\sqrt{3} \), \( a_3 = 54 \), \( a_4 = 64\sqrt{5} \), \( a_5 = 125\sqrt{6} \).

Step 1: Define the Sequence Formula

The general formula for the sequence is given as \( a_{n} = n^{3} \sqrt{n+1} \). We need to apply this formula to find the values of \( a_{n} \) from \( n=0 \) to \( n=5 \).

Step 2: Calculate \( a_{0} \)

Substitute \( n=0 \) into the sequence formula: \( a_{0} = 0^{3} \sqrt{0+1} = 0 \cdot 1 = 0 \). Thus, \( a_{0} = 0 \).

Step 3: Calculate \( a_{1} \)

Substitute \( n=1 \) into the sequence formula: \( a_{1} = 1^{3} \sqrt{1+1} = 1 \cdot \sqrt{2} = \sqrt{2} \). So, \( a_{1} = \sqrt{2} \).

Step 4: Calculate \( a_{2} \)

Substitute \( n=2 \) into the sequence formula: \( a_{2} = 2^{3} \sqrt{2+1} = 8 \cdot \sqrt{3} = 8\sqrt{3} \). Thus, \( a_{2} = 8\sqrt{3} \).

Step 5: Calculate \( a_{3} \)

Substitute \( n=3 \) into the sequence formula: \( a_{3} = 3^{3} \sqrt{3+1} = 27 \cdot \sqrt{4} = 27 \cdot 2 = 54 \). So, \( a_{3} = 54 \).

Step 6: Calculate \( a_{4} \)

Substitute \( n=4 \) into the sequence formula: \( a_{4} = 4^{3} \sqrt{4+1} = 64 \cdot \sqrt{5} = 64\sqrt{5} \). Thus, \( a_{4} = 64\sqrt{5} \).

Step 7: Calculate \( a_{5} \)

Substitute \( n=5 \) into the sequence formula: \( a_{5} = 5^{3} \sqrt{5+1} = 125 \cdot \sqrt{6} = 125\sqrt{6} \). So, \( a_{5} = 125\sqrt{6} \).

Key Concepts

Mathematical Sequences

A mathematical sequence is an ordered list of numbers, often following a specific rule or pattern. Sequences can number from finite to infinite. They vary from arithmetic sequences, where each term is a certain amount larger than the previous one, to geometric sequences, where each term is multiplied by a fixed number. In this exercise, our sequence is defined by the formula \(a_{n}=n^3 \sqrt{n+1}\). This means each term is calculated by cubing the term position \(n\) and multiplying it by the square root of \(n+1\).

Calculating terms from a sequence is simply about substituting values into the sequence formula. For example, for \(n=0\) in our sequence, replace \(n\) with 0 in the formula to find \(a_0\). Understanding the sequence formula is key to unlocking all terms in this ordered list.

Calculus Applications

Calculus, a branch of mathematics, deals extensively with change and motion. It's often used to analyze sequences to understand their behavior as the terms increase indefinitely. Calculus can ascertain limits, differentiate sequence functions, or check for convergence. For instance, our sequence \(a_{n}=n^3 \sqrt{n+1}\) increases rapidly as \(n\) grows.

To further explore, the derivative of the sequence formula might be computed to discover rates of change or assess if there's a turning point. In more advanced applications, understanding such sequences' behavior can help in fields like physics or economics, where these increasing patterns are often analyzed.

Problem Solving in Mathematics

Mathematical problem solving is an essential skill. It starts with understanding the problem, identifying knowns and unknowns, and selecting appropriate strategies or calculations to find a solution. In sequence analysis, problem solving often involves substituting values into a sequence formula to find specific terms.

For our specific problem, the approach was straightforward yet systematic. Each step involved substituting the given \(n\) values from 0 through 5 into the sequence formula, easing the process through a clear, repeatable pattern. This approach in mathematical problem solving promotes accuracy, enhances comprehension, and builds a strong foundation for tackling more complex problems.