Problem 2 - Precalculus Mathematics for Calculus

Question

The sequence \(a_{n}=a+(n-1) d\) is an arithmetic sequence in which \(a\) is the first term and \(d\) is the ____ So for the arithmetic sequence \(a_{n}=2+5(n-1)\) the first term is____ and the common difference is _____.

Step-by-Step Solution

Answer

The first term is 2, and the common difference is 5.

Step 1: Identify the Given Arithmetic Sequence Formula

We are given the arithmetic sequence formula as \( a_{n} = 2 + 5(n-1) \). This formula follows the pattern \( a_{n} = a + (n-1) d \).

Step 2: Extract the First Term

From the formula \( a_{n} = 2 + 5(n-1) \), compare it to \( a_{n} = a + (n-1)d \), you can see that the first term \( a \) is \( 2 \).

Step 3: Determine the Common Difference

In the given formula \( a_{n} = 2 + 5(n-1) \), the term \( 5(n-1) \) indicates that the common difference \( d \) is \( 5 \).

Key Concepts

First Term

In an arithmetic sequence, the first term, denoted by \( a \), is the starting point of the sequence. It represents the initial value from which the sequence progresses. Understanding what this term means is crucial because it sets the foundation for the entire sequence to unfold. Let's take a look at the formula for an arithmetic sequence: \[ a_{n} = a + (n-1) d \] Here, \( a \) is the first term. It is what you would get when you substitute 1 for \( n \) in the formula. In the example given, \( a_{n} = 2 + 5(n-1) \), you can clearly note that \( a = 2 \). This means the sequence starts with the number 2.

• The first term is essential as it establishes the beginning of the pattern.
• It helps determine subsequent terms when combined with the common difference.

Remember, every arithmetic sequence will have its distinct first term, setting it apart from others.

Common Difference

The common difference, often represented by \( d \), is a key component that determines how the sequence progresses from one term to the next. It defines the numerical gap between each consecutive term in the sequence.When looking at the formula for an arithmetic sequence, \( a_{n} = a + (n-1) d \), the common difference \( d \) is the coefficient of the \( (n-1) \) term. In our specific example \( a_{n} = 2 + 5(n-1) \), the common difference is \( 5 \). This means that each term in the sequence increases by 5 from the previous term.

• The common difference dictates the pace of growth or decline in the sequence.
• It is constant throughout the sequence, meaning every step up or down is uniform.
• A positive \( d \) results in an increasing sequence, while a negative \( d \) leads to a decreasing one.

Knowing the common difference helps you quickly predict other terms without having to manually calculate each one step by step.

Sequence Formula

The sequence formula for arithmetic sequences is a powerful tool that helps in understanding and predicting the order of terms. This formula is: \[ a_{n} = a + (n-1)d \] Each part of this formula has a specific purpose:

• \( a_{n} \): Represents any term in the sequence, where \( n \) is the position of the term.
• \( a \): The first term, serving as the base value from which others are derived.
• \( (n-1) \): Adjusts the index to account for movement along the sequence.
• \( d \): The common difference, ensuring a consistent increase or decrease between terms.

The sequence formula simplifies potentially complex calculations. For instance, if you needed to find the 6th term of the sequence \( a_{n} = 2 + 5(n-1) \), simply substitute \( n = 6 \) into the formula: \[ a_6 = 2 + 5(6-1) = 2 + 25 = 27 \]This ability to quickly determine any term makes the sequence formula indispensable for those studying arithmetic sequences.