Problem 35 - Precalculus: Mathematical for Calculus

Question

The 100 th term of an arithmetic sequence is \(98,\) and the common difference is \(2 .\) Find the first three terms.

Step-by-Step Solution

Answer

The first three terms are -100, -98, and -96.

Step 1: Understand the Problem

We need to find the first three terms of an arithmetic sequence where the 100th term is 98 and the common difference is 2.

Step 2: Recall the Formula for the nth Term of an Arithmetic Sequence

The formula to find the nth term of an arithmetic sequence is given by \( a_n = a_1 + (n-1)d \), where \( a_n \) is the nth term, \( a_1 \) is the first term, \( n \) is the term number, and \( d \) is the common difference.

Step 3: Set Up the Equation for the 100th Term

Using the formula \( a_n = a_1 + (n-1)d \), substitute \( a_{100} = 98 \), \( n = 100 \), and \( d = 2 \). The equation becomes: \[ 98 = a_1 + (100-1) imes 2 \] which simplifies to: \[ 98 = a_1 + 198 \]

Step 4: Solve for the First Term

Rearrange the equation \( 98 = a_1 + 198 \) to solve for \( a_1 \): \[ a_1 = 98 - 198 \] \[ a_1 = -100 \] Thus, the first term \( a_1 \) is \(-100\).

Step 5: Find the First Three Terms Using the First Term and the Common Difference

Use the first term \( a_1 = -100 \) and the common difference \( d = 2 \) to find the first three terms: - First term: \( a_1 = -100 \) - Second term: \( a_2 = a_1 + d = -100 + 2 = -98 \) - Third term: \( a_3 = a_1 + 2d = -100 + 2 \times 2 = -96 \) Therefore, the first three terms are \(-100\), \(-98\), and \(-96\).

Key Concepts

Common Difference

In an arithmetic sequence, the common difference is a crucial element. It is the amount by which we increase or decrease each term to obtain the next term in the sequence. If you have two consecutive terms, you can calculate the common difference by subtracting the first term from the second term.

For example, if the terms are 3 and 5, the common difference is \(5 - 3 = 2\). When this difference is consistent throughout the sequence, it indicates an arithmetic sequence.

• If the difference is positive, the sequence increases.
• If the difference is negative, the sequence decreases.

The understanding of common difference helps in predicting and forming future sequence terms, ensuring you can easily generate any number of terms required.

nth Term Formula

The nth term formula is a powerful tool to find any specific term of an arithmetic sequence. This formula allows you to determine the value of the term in the sequence without listing all the previous terms. The formula is:
\[ a_n = a_1 + (n-1)d \]
where:

• \(a_n\) is the nth term you want to find,
• \(a_1\) is the first term of the sequence,
• \(n\) is the term position, and
• \(d\) is the common difference.

To use this formula, simply plug in the values you know. For instance, to find the 100th term of a sequence where the first term is \(-100\) and the common difference is \(2\), the application looks like:
\[ a_{100} = -100 + (100 - 1) \times 2 = -100 + 198 = 98 \].
This formula is efficient and essential for quick calculations without manual listings.

Sequence Terms

Finding the sequence terms in arithmetic sequences is straightforward once you know the first term and the common difference. It essentially involves adding the common difference to the term repeatedly.

Here's how you find the first few terms, say, the first three terms, based on known information:

• Start with the first term: For an arithmetic sequence with the first term \(a_1 = -100\), this term is the sequence's beginning.
• Second term: Add the common difference to the first term. So, \(a_2 = a_1 + d = -100 + 2 = -98\).
• Third term: Add the common difference to the second term. Consequently, \(a_3 = a_2 + d = -98 + 2 = -96\).

By understanding and applying these basics, you can efficiently compute any sequence term with little effort.