Problem 32 - Algebra for College Students

Question

Write a formula for the general term of each infinite sequence. \(4,6,8,10,12, \dots\)

Step-by-Step Solution

Answer

The general term of the sequence is \( a_n = 2n + 2 \).

Step 1: Identify the pattern in the sequence

The given sequence is: 4, 6, 8, 10, 12, ... Observe how each term transforms into the next one. Notice that each term is 2 more than the previous term.

Step 2: Determine the first term and common difference

The first term of the sequence, denoted as \(a_1\), is 4. The difference between consecutive terms, known as the common difference \(d\), is 2.

Step 3: Use the arithmetic sequence formula

The general formula for the nth term of an arithmetic sequence is: \( a_n = a_1 + (n-1) \times d \)

Step 4: Substitute the values into the formula

Using \(a_1 = 4\) and \(d = 2\), substitute these into the arithmetic sequence formula to get: \( a_n = 4 + (n-1) \times 2 \)

Step 5: Simplify the formula

Expand and simplify the expression: \( a_n = 4 + 2n - 2\) Combine like terms to get \( a_n = 2n + 2 \)

Key Concepts

General Term Formula

In any arithmetic sequence, we can find a specific term by using a general term formula. The formula helps us identify the value of any term without listing all previous terms. For an arithmetic sequence, the general term formula is given by \( a_n = a_1 + (n-1) \times d \) Here, \( a_n \) denotes the nth term, \( a_1 \) is the first term, \( n \) is the position of the term in the sequence, and \( d \) represents the common difference. Using this formula makes it easy to calculate any desired term within an arithmetic sequence efficiently.

Common Difference

The common difference in an arithmetic sequence is the consistent amount added (or subtracted) to each term to get the next term. You can find it by subtracting any term from the term that follows it. In our example sequence: \( 4, 6, 8, 10, 12, \ldots \) we see that each term is 2 more than the previous term. Thus, \( d = 6 - 4 = 2 \) Repeating this process between multiple terms shows that the common difference remains 2 throughout. Recognizing the common difference is crucial for using the general term formula correctly.

Nth Term

The nth term of an arithmetic sequence refers to the term located at position \( n \). It's crucial because it allows us to calculate any term's value directly without generating all the preceding terms. From the given example, we know the first term \( a_1 \) is 4 and the common difference \( d \) is 2. Using our general term formula: \( a_n = a_1 + (n-1) \times d \) we substitute the known values: \( a_n = 4 + (n-1) \times 2 \) Simplify to: \( a_n = 4 + 2n - 2 \) Combine like terms to get: \( a_n = 2n + 2 \) This tells us that the value of any term \( a_n \) in our sequence can be quickly calculated using the formula \( 2n + 2 \).