Problem 35 - Precalculus with Limits

Question

In Exercises 33 - 40, write the first five terms of the arithmetic sequence. \( a_1 = -2.6, d = -0.4 \)

Step-by-Step Solution

Answer

The first five terms of the sequence are: -2.6, -3.0, -3.4, -3.8, -4.2.

Step 1: Calculating the First Term

The first term (\( a_1 \)) is already given as -2.6.

Step 2: Calculating the Second Term

Utilize the formula for the nth term of an arithmetic sequence: \( a_n = a_1 + (n-1) * d \). Here, n=2, \( a_1 = -2.6 \), and d=-0.4. So, \( a_2 = -2.6 + (2-1)* -0.4 = -3.0 \).

Step 3: Calculating the Third Term

Following the same process, substitute n=3 to find \( a_3 \). Hence, \( a_3 = -2.6 + (3-1)*-0.4 = -3.4 \) .

Step 4: Calculating the Fourth Term

The fourth term, similarly, is \( a_4 = -2.6 + (4-1)*-0.4 = -3.8 \).

Step 5: Calculating the Fifth Term

Finally, for \( a_5 = -2.6 + (5-1)*-0.4 \), which gives us -4.2.

Key Concepts

Arithmetic Progression

Understanding an arithmetic progression (AP) is key to maneuvering through a range of mathematical problems. It is a sequence of numbers in which each term after the first is found by adding a constant, known as the common difference, to the preceding term. This can be easily visualized with examples like 2, 4, 6, 8, where the common difference is 2.

In the given exercise, the sequence starts with a term of -2.6, and the common difference is -0.4. This negative common difference indicates that the sequence is decreasing, moving towards more negative values with each succeeding term.

Sequence and Series

A sequence is an ordered list of numbers following a specific pattern, while a series refers to the sum of these numbers. Sequences can be finite or infinite depending on whether they have an end or continue indefinitely. Arithmetic progression is a type of sequence wherein the difference between consecutive terms is constant. Meanwhile, the concept of a series is essential when working with arithmetic sequences if the goal is to sum the terms, for instance finding the sum of the first five terms. Luckily, in our exercise, the focus is on identifying the individual terms, not their sum.

Nth Term Formula

The nth term formula is a magical tool in the world of sequences, providing a direct way to find any term in an arithmetic sequence without needing to list out all the previous ones. It's given by the formula:
\( a_n = a_1 + (n - 1)d \)
where \(a_n\) is the nth term, \(a_1\) is the first term, n is the term number, and d is the common difference. In the context of our exercise, using this formula allowed us to methodically calculate each term up to the fifth one, simply by plugging in the values for n and the given common difference, demonstrating the formula's power in simplifying the process.