Problem 66 - Algebra 2

Question

Find the indicated term of each arithmetic sequence. $$ a_{1}=46, d=5, n=14 $$

Step-by-Step Solution

Answer

The 14th term is 111.

Step 1: Identify the Formula for the nth Term

For an arithmetic sequence, the formula to find the nth term is given by \( a_n = a_1 + (n-1) \cdot d \), where \( a_1 \) is the first term, \( d \) is the common difference, and \( n \) is the term number we want to find. In this problem: \( a_1 = 46 \), \( d = 5 \), and \( n = 14 \).

Step 2: Substitute Known Values into Formula

Substitute the given values into the formula: \( a_{14} = 46 + (14-1) \cdot 5 \). Simplify the expression in the parentheses first.

Step 3: Simplify the Expression

Calculate the value of \( (14-1) \) which is 13. This gives us: \( a_{14} = 46 + 13 \cdot 5 \).

Step 4: Perform Multiplication and Add

Multiply 13 by 5 to get 65. Now, add 65 to 46 to find \( a_{14} \). So, \( a_{14} = 46 + 65 = 111 \).

Key Concepts

nth term formula

In the world of arithmetic sequences, the "nth term formula" is your best friend when it comes to predicting and calculating specific terms within the sequence. The formula is straightforward: \( a_n = a_1 + (n-1) \cdot d \). Here's what each part means:

• \( a_n \) represents the "nth" term that you're trying to find.
• \( a_1 \) is the first term in the sequence. It's where it all begins.
• \( d \) is the common difference, or the consistent step you add to each term to get to the next.
• \( n \) is the number of the term you want to find.

Understanding this formula allows you to dive into a sequence at any point, without having to manually calculate every preceding term. Just slot in your values and solve. For example, if you wanted the 14th term in a sequence starting at 46 with a step of 5, you'd use these parameters: \( a_1 = 46 \), \( d = 5 \), and \( n = 14 \). Follow the formula, and finding the term becomes just simple arithmetic.

common difference

The "common difference" in an arithmetic sequence is like the beat of a drum—it keeps everything in rhythm and predictable. This is the number you repeatedly add to each term to get to the next one. It's symbolized by \( d \) in our formula.Let's highlight some key points:

• The common difference is what makes an arithmetic sequence linear and predictable. Each step is even and gives the sequence its unique identity.
• A positive \( d \) makes the sequence increase, while a negative \( d \) makes it decrease.
• If \( d = 0 \), every term in the sequence is the same, essentially a constant sequence.

To see it in action: In our example problem, \( d = 5 \). Starting from 46, every time you add 5 to get the next number, you're producing another term in your sequence.

sequence terms

When exploring "sequence terms" in arithmetic sequences, think of each term as a stepping stone along a path. Each term is a result of the first term and the consistent application of the common difference. Here’s how they form the backbone of sequences:The first term, symbolized by \( a_1 \), initiates the sequence. From there, each subsequent term (second, third, and so on) is calculated with the formula \( a_n = a_1 + (n-1) \cdot d \).Key insights include:

• The sequence terms are what you see when you list the numbers: 46, 51, 56, 61, and so forth in our example.
• Understanding how to calculate any term means you can identify patterns, predict future terms, and even solve for unknowns.

So with the given \( a_1 = 46 \) and \( d = 5 \), the terms of our sequence rise steadily by 5, leading up to the 14th term we've calculated as 111. This constant progression is the heart of what makes arithmetic sequences so intriguing and useful.