Problem 28 - Precalculus with Limits

Question

In Exercises 9-32, write the first five terms of the sequence. (Assume that \( n \) begins with 1.) \( a_n = 0.3 \)

Step-by-Step Solution

Answer

The first five terms of the sequence \( a_n = 0.3 \) are: 0.3, 0.3, 0.3, 0.3, 0.3.

Step 1: Substitute n with 1.

To find the first term, substitute n = 1 into the expression: \( a_1 = 0.3 \).

Step 2: Substitute n with 2.

To find the second term, substitute n = 2 into the expression: \( a_2 = 0.3 \).

Step 3: Substitute n with 3.

To find the third term, substitute n = 3 into the expression: \( a_3 = 0.3 \).

Step 4: Substitute n with 4.

To find the fourth term, substitute n = 4 into the expression: \( a_4 = 0.3 \)

Step 5: Substitute n with 5.

To find the fifth term, substitute n = 5 into the expression: \( a_5 = 0.3 \).

Key Concepts

Understanding Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This difference is known as the "common difference." For example, in the sequence 3, 5, 7, 9, ..., the common difference is 2 since each term increases by 2 from the previous one.

To identify an arithmetic sequence, look for sequences where this pattern of adding or subtracting the same number repeatedly emerges. If every subsequent number is acquired by adding or subtracting a fixed number from the previous term, then it is likely an arithmetic sequence.

• Example: 12, 14, 16, 18: This is an arithmetic sequence with a common difference of 2.
• Non-Example: 1, 2, 4, 8: This sequence does not have a constant difference, thus, it is not arithmetic.

Recognizing arithmetic sequences is crucial because they often appear in algebra and problem-solving scenarios. Once identified, arithmetic sequences can make calculations predictable and manageable.

Exploring Sequence Terms

Sequence terms are the individual elements in a sequence, typically denoted by a subscript. In the sequence provided, each term is symbolized as \( a_n \), where \( n \) serves as the position number of the term. For example, \( a_1 \) represents the first term, \( a_2 \) the second, and so on. Being familiar with sequence terms is fundamental in mathematics as they help represent and solve problems efficiently.

• A term's position gives its label, like \( a_1 = 0.3 \), \( a_2 = 0.3 \), etc.
• In a sequence, each term is influenced directly by its predecessor in many cases, especially in arithmetic sequences.

The clarity of understanding sequence terms helps in determining patterns or formulas governing a sequence. Students frequently deal with sequence terms in subjects such as algebra and calculus, where they might need to express or evaluate sequential progressions.

Characteristics of a Constant Sequence

A constant sequence is a unique type of sequence where all the terms are identical. This means that no matter how far along the sequence you go, each of its terms has the same value. The sequence in the exercise, \( a_n = 0.3 \), is a constant sequence since every term equals 0.3.

• In a constant sequence, the common difference is zero because there is no change between consecutive terms.
• Example: 5, 5, 5, 5, ... – Here, every term is the same, fulfilling the definition of a constant sequence.

Constant sequences are easy to work with because they simplify calculations and equations, often providing stability in analyzing patterns or solving equations. This simplicity is helpful in arithmetic progressions involving specific scenarios like amortization schedules in finance or when modeling unchanging conditions in science.