Problem 20 - Algebra and Trigonometry

Question

17-22 Find the first five terms of the sequence and determine if it is arithmetic. If is arithmetic, find the common difference and express the \(n\)th term of the sequence in the standard form \(a_{n}=a+(n-1) d .\) $$a_{n}=1+\frac{n}{2}$$

Step-by-Step Solution

Answer

The sequence is arithmetic with a common difference of 0.5; \( a_n = 0.5n + 1 \).

Step 1: Generate the first term

To find the first term of the sequence, substitute \( n = 1 \) into the expression for \( a_n \). Thus, \( a_1 = 1 + \frac{1}{2} = 1.5 \).

Step 2: Generate the second term

Substitute \( n = 2 \) into the expression for \( a_n \). Hence, \( a_2 = 1 + \frac{2}{2} = 2 \).

Step 3: Generate the third term

Substitute \( n = 3 \) into the expression for \( a_n \). Thus, \( a_3 = 1 + \frac{3}{2} = 2.5 \).

Step 4: Generate the fourth term

Substitute \( n = 4 \) into the expression for \( a_n \). Hence, \( a_4 = 1 + \frac{4}{2} = 3 \).

Step 5: Generate the fifth term

Substitute \( n = 5 \) into the expression for \( a_n \). Thus, \( a_5 = 1 + \frac{5}{2} = 3.5 \).

Step 6: Determine if the sequence is arithmetic

An arithmetic sequence has a constant difference between consecutive terms. Calculate the differences: \( 2 - 1.5 = 0.5 \), \( 2.5 - 2 = 0.5 \), \( 3 - 2.5 = 0.5 \), \( 3.5 - 3 = 0.5 \). Since all differences are the same, the sequence is arithmetic.

Step 7: Express the nth term in standard form

The first term \( a_1 \) is 1.5, and the common difference \( d \) is 0.5. The nth term of an arithmetic sequence is given by \( a_n = a_1 + (n-1) \cdot d \). Thus, \( a_n = 1.5 + (n-1) \cdot 0.5 \). Simplifying gives \( a_n = 0.5n + 1 \).

Key Concepts

Common Difference

The common difference is a critical part of understanding arithmetic sequences. It is the constant difference between consecutive terms in a sequence. In simpler terms, it's what you add to each term to get to the next one.
For example, in the sequence we are studying:

• The first term is 1.5, the second term is 2, and so on: 1.5, 2, 2.5, 3, 3.5...
• Finding the common difference involves subtracting any term from its succeeding term. In our sequence: 2 - 1.5 = 0.5, 2.5 - 2 = 0.5, 3 - 2.5 = 0.5, and 3.5 - 3 = 0.5.

The result is always 0.5 here, confirming the sequence has a common difference.
This constant nature of the common difference is what characterizes a sequence as arithmetic. Once you find this stable pattern, it becomes straightforward to predict the following terms or set up a general formula.

nth Term Formula

A central feature of arithmetic sequences is the ability to express them with an nth term formula. This formula enables us to calculate any term in the sequence without listing all preceding terms.
The standard equation for the nth term of an arithmetic sequence is:
\( a_n = a_1 + (n-1) \cdot d \)
Where:

• \( a_n \) is the nth term,
• \( a_1 \) is the first term of the sequence,
• \( d \) is the common difference, and
• \( n \) is the term number.

In our exercise, the first term, \( a_1 \), is 1.5 and the common difference, \( d \), is 0.5.
Thus, the nth term formula becomes:\[ a_n = 1.5 + (n-1) \cdot 0.5 \]
Simplifying, we achieve:\[ a_n = 0.5n + 1 \]This equation is concise and practical, allowing us to find any term easily by substituting the desired term number in place of \( n \). For example, to find the tenth term, we'd compute \( a_{10} = 0.5 \times 10 + 1 = 6 \).

Sequence Analysis

Analyzing a sequence helps us understand its properties and behavior. When exploring sequences, especially arithmetic ones, there are several key aspects to focus on.
Here’s what we usually analyze:

• Classification: Recognizing what type of sequence it is by looking at differences. Here, spotting a constant difference of 0.5 between terms identifies it as arithmetic.
• Pattern Recognition: This involves understanding the repetitive nature or pattern. For instance, every term steadily increases by 0.5.
• Constructing a Formula: Developing the nth term formula, as discussed earlier, can showcase how each element in the sequence relates to its position. This formula aids in calculating any term efficiently.
• Future Predictions: With the nth term formula, we can predict not just the immediate next term, but any term far into the sequence, like the 100th term!

Engaging in sequence analysis not only helps in solving textbook problems but also encourages logical thinking. Moreover, it strengthens understanding of how simple arithmetic rules can apply to numerous scenarios.