Problem 27 - Algebra 2

Question

Find the indicated term of each arithmetic sequence. \(a_{1}=3, d=7, n=14\)

Step-by-Step Solution

Answer

The 14th term is 94.

Step 1: Understand the Formula

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

Step 2: Identify Known Values

From the exercise, we know: \( a_1 = 3 \), \( d = 7 \), and \( n = 14 \). We'll use these values in our formula.

Step 3: Plug Values into the Formula

Substitute the known values into the formula: \[ a_{14} = 3 + (14-1) \times 7 \]

Step 4: Simplify the Expression

Calculate inside the parentheses first: \( 14-1 = 13 \).Now the expression is: \[ a_{14} = 3 + 13 \times 7 \]

Step 5: Perform Multiplication

Multiply the common difference by \( 13 \): \( 13 \times 7 = 91 \).The expression now is: \[ a_{14} = 3 + 91 \]

Step 6: Calculate the Final Sum

Add \( 3 \) and \( 91 \) to get the final answer: \[ a_{14} = 3 + 91 = 94 \]

Key Concepts

n-th term

In an arithmetic sequence, each term is found by adding a constant value, known as the common difference, to the previous term. The term you want to find in an arithmetic sequence is called the n-th term. The position of this term is represented by the letter "n". The formula to find the n-th term is \( a_n = a_1 + (n-1) \times d \). Here, \( a_n \) represents the n-th term, \( a_1 \) is the first term in the sequence, \( d \) is the common difference, and \( n \) is the term number.

Let's break down this formula:

• \( a_1 \) represents where the sequence begins, effectively setting the stage for all subsequent numbers.
• \( (n-1) \) gives you how many times you need to add the common difference \( d \) to reach the n-th term.
• The product \( (n-1) \times d \) calculates how far you need to move from the first term to get to the n-th term.

By substituting these known values into the formula, you can easily calculate any term in an arithmetic sequence.

common difference

The backbone of an arithmetic sequence is the common difference. This constant, denoted by \( d \), determines how much each term increases or decreases from the previous one.
This sequence is consistent, which means each term is found by simply adding or subtracting this difference, which makes calculations and predictions straightforward.
For example, in our problem, the common difference is \( 7 \).

• To find each subsequent term, you add \( 7 \) to the previous term.
• If the common difference were negative, you'd subtract instead, which would result in a decreasing sequence.
• The constant nature of the common difference ensures that the sequence's pattern never changes, making it easy to compute distant terms using the formula.

Knowing the common difference in an arithmetic sequence empowers you to understand how the sequence progresses and enables you to find any term.

first term

The first term, often denoted as \( a_1 \), sets the starting point for an arithmetic sequence. It significantly impacts the rest of the sequence since every term is calculated based on this initial value.
In our exercise, the first term \( a_1 \) is given as \( 3 \).

• It's the term where your sequence begins; without it, you cannot properly define the sequence.
• Every subsequent term is derived by repeatedly adding the common difference \( d \) to this first term.
• In essence, the first term anchors the sequence, and its value is crucial to not just determining the n-th term, but also understanding the overall nature and behavior of the sequence.

The first term is essential because it entirely determines the position and values of all other terms when paired with the common difference.