Problem 11 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

The values are \(0, \frac{1}{2}, \frac{4}{3}, \frac{9}{4}, \frac{16}{5}, \frac{25}{6}\).

Step 1: Understanding the Sequence Formula

The sequence is given by the formula \(a_n = \frac{n^2}{n+1}\). We need to use this formula to calculate each term of the sequence for \(n=0, 1, 2, 3, 4,\) and \(5\).

Step 2: Calculate \(a_0\)

Substitute \(n = 0\) into the formula: \(a_0 = \frac{0^2}{0+1} = \frac{0}{1} = 0\).

Step 3: Calculate \(a_1\)

Substitute \(n = 1\) into the formula: \(a_1 = \frac{1^2}{1+1} = \frac{1}{2}\).

Step 4: Calculate \(a_2\)

Substitute \(n = 2\) into the formula: \(a_2 = \frac{2^2}{2+1} = \frac{4}{3}\).

Step 5: Calculate \(a_3\)

Substitute \(n = 3\) into the formula: \(a_3 = \frac{3^2}{3+1} = \frac{9}{4}\).

Step 6: Calculate \(a_4\)

Substitute \(n = 4\) into the formula: \(a_4 = \frac{4^2}{4+1} = \frac{16}{5}\).

Step 7: Calculate \(a_5\)

Substitute \(n = 5\) into the formula: \(a_5 = \frac{5^2}{5+1} = \frac{25}{6}\).

Key Concepts

Understanding Arithmetic Sequences

Arithmetic sequences are a type of number pattern where each term after the first is the sum of the previous one and a constant difference. This difference is called the "common difference." For example, in the sequence 2, 4, 6, 8, the common difference is 2.In an arithmetic sequence, you can find any term using the formula:\[ a_n = a_1 + (n-1) \cdot d \]where \(a_1\) is the first term, \(n\) is the term number, and \(d\) is the common difference. However, not all sequences are arithmetic. In the sequence from the exercise, each term is derived from a specific formula: \[ a_n = \frac{n^2}{n+1} \]This formula indicates that as each term increases in value based on the relationship between the numerator \(n^2\) and denominator \(n+1\). This is not an arithmetic sequence, since there is no constant difference between terms.

Using Mathematical Formulas

Mathematical formulas are tools used to solve problems in a structured way. They give us a method to calculate unknown values based on known quantities. In the sequence calculation example, the formula provided is:\[ a_n = \frac{n^2}{n+1} \]This specific formula uses division and powers to define each term in the sequence based on the variable \(n\). For every increment in \(n\), you substitute the new value into the formula to calculate each subsequent term. Understanding how to manipulate and substitute values into such formulas is crucial for problem-solving in mathematics, as it enables the precise computation of results.

Step-by-Step Solutions Approach

A step-by-step solution helps break down a problem into manageable parts, making it easier to comprehend and solve. Here's how you can approach such problems efficiently:

• **Identify the Formula:** Start by clearly understanding and writing down the formula you'd use, like in our problem \(a_n = \frac{n^2}{n+1}\).
• **Substitute Values:** For each calculation, replace \(n\) with the specific value required, such as \(n = 0, 1, 2, ..., 5\).
• **Simplify and Solve:** Perform any arithmetic operations needed to simplify the expression and find the answer for each term.

These steps, when followed systematically, ensure you don't miss any critical points and help deliver an accurate solution for each term in a series of calculations. Writing things out explicitly often makes it easier to spot errors or areas of confusion.