Problem 6 - Precalculus Mathematics for Calculus

Question

Find the first four terms and the 100th term of the sequence. $$a_{n}=n^{2}+1$$

Step-by-Step Solution

Answer

The first four terms are 2, 5, 10, 17, and the 100th term is 10001.

Step 1: Understand the Sequence Formula

The given sequence formula is \(a_n = n^2 + 1\). This means that for any term \(a_n\), you substitute \(n\) with the term number in the formula.

Step 2: Calculate the First Term

Substitute \(n = 1\) into the sequence formula: \(a_1 = 1^2 + 1 = 2\). Therefore, the first term is 2.

Step 3: Calculate the Second Term

Substitute \(n = 2\) into the sequence formula: \(a_2 = 2^2 + 1 = 5\). Therefore, the second term is 5.

Step 4: Calculate the Third Term

Substitute \(n = 3\) into the sequence formula: \(a_3 = 3^2 + 1 = 10\). Therefore, the third term is 10.

Step 5: Calculate the Fourth Term

Substitute \(n = 4\) into the sequence formula: \(a_4 = 4^2 + 1 = 17\). Therefore, the fourth term is 17.

Step 6: Calculate the 100th Term

Substitute \(n = 100\) into the sequence formula: \(a_{100} = 100^2 + 1 = 10001\). Therefore, the 100th term is 10001.

Key Concepts

Term Calculation

Term calculation in sequences involves finding specific terms based on the sequence formula provided. In our given example, the formula is \(a_n = n^2 + 1\). To find a particular term in this sequence, you need to substitute the term number into the formula. For example, if you want to find the first term, you set \(n = 1\). This results in calculating \(a_1 = 1^2 + 1 = 2\), which means the first term is 2.

The process is similar for any term in the sequence. Just substitute the term number into the formula and perform the calculation:

• Second Term: Set \(n = 2\), calculate \(a_2 = 2^2 + 1 = 5\).
• Third Term: Set \(n = 3\), calculate \(a_3 = 3^2 + 1 = 10\).
• Fourth Term: Set \(n = 4\), calculate \(a_4 = 4^2 + 1 = 17\).
• 100th Term: Set \(n = 100\), calculate \(a_{100} = 100^2 + 1 = 10001\).

Calculating terms helps you see specific values within the sequence, which is especially useful for understanding how sequences grow or change as they progress.

Sequence Formula

The sequence formula is crucial as it defines how each term in a sequence is generated based on its position. In our sequence, the formula is \(a_n = n^2 + 1\). Each new value of \(n\) represents the position in the sequence and thus determines the value of \(a_n\). This formula acts as a blueprint for constructing each term.

Understanding the sequence formula is essential because:

• It provides a systematic way to determine terms without manually counting or listing them.
• It helps identify the pattern or rule that the sequence follows, which can reveal deeper insights about its nature.
• A clear sequence formula can simplify calculations, making it easier to find terms that are far into the sequence, such as the 100th term.

A sequence formula like \(a_n = n^2 + 1\) is simple yet powerful, providing a direct calculation method for any term, thereby illustrating the relationship between term numbers and their values.

Substitution in Sequences

Substitution in sequences is the method of replacing the term number, represented by \(n\), with specific numbers to find the value of a given term. This step is crucial because it applies the sequence formula practically, allowing us to extract exact values for terms.

Here's how substitution works in this context:

• Pick a term number: Choose the position number, such as 1, 2, 3, etc.
• Replace \(n\) in the formula \(a_n = n^2 + 1\) with the chosen number.
• Perform the calculation: Execute the arithmetic to find the term, for example, \(a_1 = 1^2 + 1\).

This process of substitution provides a straightforward technique for deriving terms. Whether you're looking for the initial terms or a distant one like the 100th, substitution in sequences simplifies the work by applying a consistent mathematical approach.