Problem 67 - Algebra 2

Question

Find the 27 th term of each sequence. $$ -11,-5,1, \ldots $$

Step-by-Step Solution

Answer

The 27th term of the sequence is 145.

Step 1: Identify the first term and the common difference

The first term \( a_1 \) is -11, and the common difference \( d \) can be calculated as the difference between any second term and the first preceding term. For instance, -5 - (-11) = 6, or 1 - (-5) = 6.

Step 2: Use the general term formula

The sequence follows an arithmetic progression, so the general formula for finding the nth term of an arithmetic sequence is: \( a_n = a_1 + (n - 1) * d \)

Step 3: Substitute and calculate

Substitute \( a_1 \) with -11, \( n \) with 27 (because we're looking for the 27th term), and \( d \) with 6 into the formula and evaluate the result: \( a_n = -11 + (27 - 1) * 6 \)

Key Concepts

Common Difference

To really understand an arithmetic sequence, it's essential to know what the **common difference** is. This term is used to define the consistent change between every consecutive pair of terms in the sequence. For example, in the sequence given in the problem (-11, -5, 1,...), the common difference is 6.

You can find the common difference by subtracting any term from the term that follows it. In our sequence:

• -5 - (-11) = 6
• 1 - (-5) = 6

This value doesn't change throughout the sequence, which is why it's called "common." It's the "step size" you move by to get from one term to the next in an arithmetic progression. Knowing the common difference is crucial because it helps you find other terms in the sequence using the **general term formula**.

General Term Formula

The **general term formula** is like a magical tool that lets you find any term in an arithmetic sequence. This formula is:

• \( a_n = a_1 + (n - 1) \times d \)

Here, \( a_n \) is the term you want to find, \( a_1 \) is the first term, \( n \) is the position of the term (like the 27th term), and \( d \) is the common difference. Using our sequence as an example:

• The first term \( a_1 \) is -11.
• The common difference \( d \) is 6.

To find the 27th term, plug the values into the formula:\[a_{27} = -11 + (27 - 1) \times 6\]Breaking this down step by step helps make calculations easy. Start by solving \( 27 - 1 \), which is 26, and then multiply it by the common difference of 6. Finally, add this result to the first term \( a_1 \) to find \( a_{27} \). This formula is very helpful to access any term without having to list all the prior ones.

Arithmetic Progression

An **arithmetic progression** is simply a sequence of numbers that are arranged in a particular order. Each number in this sequence is obtained by adding the common difference to the previous number. This makes it predictable and straightforward, especially for mathematical calculations.

In the example sequence -11, -5, 1, ..., each term is generated by adding the common difference of 6 to the prior term. This kind of progression is super helpful in various fields of math and real-world applications because it makes predicting future values easy.
Some features of an arithmetic progression include:

• It is linear, meaning the graph of its terms will create a straight line.
• Each new term depends on the term before it, nothing else.

By identifying a sequence as an arithmetic progression, you can use the **common difference** and **general term formula** to solve problems quickly. Whether it's finding future terms, understanding patterns, or solving real-world problems, recognizing this structured sequence is powerful.