Problem 33 - Algebra for College Students

Question

Write a formula for the general term of each infinite sequence. \(3,6,9,12, \dots\)

Step-by-Step Solution

Answer

The general term is a_n = 3n.

Step 1: Identify the Sequence Type

First, observe whether the sequence is arithmetic or geometric. Since each term increases by the same amount, this is an arithmetic sequence.

Step 2: Determine the Common Difference

Find the common difference by subtracting the first term from the second term. In this sequence, the common difference is 6 - 3 = 3.

Step 3: Write the General Formula

For an arithmetic sequence, the general term is given by a_n = a_1 + (n-1) d, where a_1 is the first term and d is the common difference. Substitute a_1 = 3 and d = 3 into the formula: a_n = 3 + (n-1)3.

Step 4: Simplify the Formula

Simplify the expression to find the final form of the general term: a_n = 3 + 3(n-1) = 3 + 3n - 3 = 3n.

Key Concepts

arithmetic sequences

An arithmetic sequence is a sequence of numbers where each term is derived by adding a constant value to the previous term. This constant value is known as the 'common difference'. For example, in the sequence 3, 6, 9, 12, ..., each term increases by 3. Understanding arithmetic sequences helps in identifying patterns and solving series problems in algebra.

Arithmetic sequences are straightforward because of their regular increments. To identify an arithmetic sequence, check if the difference between consecutive terms is consistent.

general term formula

The general term formula of an arithmetic sequence lets us find any term in the sequence without listing all previous terms. The formula is given by:
\(a_n = a_1 + (n-1) d\), where:

\(a_n\) is the nth term
\(a_1\) is the first term
\(d\) is the common difference
For the sequence 3, 6, 9, 12, ..., we have \(a_1 = 3\) and \(d = 3\). Thus, the general term formula would be:
\(a_n = 3 + (n-1) 3\)
Simplifying this, we get: \(a_n = 3n\).

This formula is crucial because it allows us to find any term in the sequence quickly. You can use this formula to identify any term's value by substituting the desired term's position (n) into the formula.

common difference

In an arithmetic sequence, the common difference is the constant amount that each term increases by. To find the common difference, subtract the first term from the second term. For the sequence 3, 6, 9, 12, ..., the common difference (d) is:
\(d = 6 - 3 = 3\)

Knowing the common difference allows you to predict the sequence and derive the general term formula. Once you have the common difference, it becomes easy to understand and work with the sequence.

The formula for the nth term, \(a_n = a_1 + (n-1) d\), depends heavily on this common difference, making it a fundamental element of arithmetic sequences.