Problem 21 - Beginning and Intermediate Algebra

Question

Find a formula for the general term, \(a_{n},\) of each sequence. $$2,4,6,8, \dots$$

Step-by-Step Solution

Answer

The formula for the general term of the given sequence is \(a_n = 2n\).

Step 1: Identify the pattern

Observe the given sequence: \(2, 4, 6, 8, \dots\). Notice that each term in the sequence is an even number and there is a constant difference of 2 between consecutive terms. This is an arithmetic sequence with a common difference, d, equal to 2.

Step 2: Identify the first term and common difference

In our observed sequence, the first term, \(a_1\), is 2 and the common difference, d, between consecutive terms is 2 as well.

Step 3: Use the arithmetic sequence formula

For an arithmetic sequence, the formula for the general term, \(a_n\), is given by: \(a_n = a_1 + (n-1)d\)

Step 4: Substitute the values of \(a_1\) and d and simplify

Now, substitute the values of \(a_1 = 2\) and \(d = 2\) into the formula and simplify: \[ a_n = 2 + 2(n-1)\newline a_n = 2 + 2n - 2\newline a_n = 2n \] So, the formula for the general term of the given sequence is \(a_n = 2n\).

Key Concepts

General Term Formula

One of the most essential things to know about arithmetic sequences is how to find the general term formula. This formula represents any term in the sequence based on its position. For arithmetic sequences, the general term formula is typically expressed as:

• \(a_n = a_1 + (n-1)d\)

Here, \(a_n\) stands for the nth term in the sequence, \(a_1\) is the first term, \(n\) is the term’s position in the sequence, and \(d\) is the common difference.
By plugging the values for \(a_1\) and \(d\) into this formula, you can calculate any term in the sequence effortlessly. It provides a powerful tool to understand and predict the progression of the sequence.

Common Difference

In arithmetic sequences, the common difference is a crucial element that links consecutive terms. It is the constant amount by which each term increases as you move through the sequence. For example, in the sequence \(2, 4, 6, 8, \ldots\), each term increases by 2.
You can calculate the common difference by subtracting one term from its subsequent term:

• \(d = a_2 - a_1\)

In our sequence, \(d = 4 - 2 = 2\). This consistency of the common difference gives arithmetic sequences their distinct linear pattern. Understanding the common difference is fundamental to applying the general term formula correctly and analyzing the sequence behavior.

Sequence Patterns

Sequence patterns help in identifying how the terms in a sequence are structured and grow. For arithmetic sequences like \(2, 4, 6, 8, \ldots\), a clear linear pattern is established by the uniform increase by the common difference.
Recognizing the patterns helps one solve, extend, or predict sequence behaviors without recalculating from the beginning each time. Patterns also simplify sequence analysis in real-world applications, like predicting future values or analyzing periodic data trends. Grasping the pattern of changes, such as consistent additions, is essential for understanding arithmetic sequences deeply.

Algebraic Formulas

Algebraic formulas are indispensable in deriving and understanding sequences. In the context of arithmetic sequences, the general term formula can be viewed as an algebraic equation. It symbolizes a linear relationship between the position of each term and its value.

• The formula \(a_n = 2n\) derived from the sequence \(2, 4, 6, 8, \ldots\) is an example of using algebraic expressions to define and calculate sequence terms efficiently

Learning to express sequences in algebraic terms equips you with the ability to manipulate and explore further mathematical sequences effortlessly. It also fosters a broader understanding of how algebra interconnects with other branches of mathematics.