Problem 18 - 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 \(a_{n}=a+(n-1) d .\) $$a_{n}=4+2^{n}$$

Step-by-Step Solution

Answer

The sequence is not arithmetic since the differences are not constant.

Step 1: Identify the First Five Terms

Let's find the first five terms of the sequence. We do this by substituting values for \( n \) from 1 to 5 into the formula \( a_{n} = 4 + 2^{n} \). - For \( n = 1 \): \( a_{1} = 4 + 2^{1} = 4 + 2 = 6 \)- For \( n = 2 \): \( a_{2} = 4 + 2^{2} = 4 + 4 = 8 \)- For \( n = 3 \): \( a_{3} = 4 + 2^{3} = 4 + 8 = 12 \)- For \( n = 4 \): \( a_{4} = 4 + 2^{4} = 4 + 16 = 20 \)- For \( n = 5 \): \( a_{5} = 4 + 2^{5} = 4 + 32 = 36 \)So, the first five terms are: 6, 8, 12, 20, 36.

Step 2: Check if the Sequence is Arithmetic

A sequence is arithmetic if the difference between consecutive terms is constant. Let's calculate these differences from our sequence:- Difference between \( a_{2} \) and \( a_{1} \): \( 8 - 6 = 2 \)- Difference between \( a_{3} \) and \( a_{2} \): \( 12 - 8 = 4 \)- Difference between \( a_{4} \) and \( a_{3} \): \( 20 - 12 = 8 \)- Difference between \( a_{5} \) and \( a_{4} \): \( 36 - 20 = 16 \)The differences are not constant (2, 4, 8, and 16), so the sequence is not arithmetic.

Step 3: Final Conclusion

Since the sequence is not arithmetic due to the variable differences between consecutive terms, we cannot find a common difference or express the \( n \)th term in the standard arithmetic sequence form \( a_{n} = a + (n-1) d \).

Key Concepts

Common Difference

An arithmetic sequence is a list of numbers where the difference between consecutive terms is always the same. This constant difference is known as the "common difference." It's a crucial element that defines an arithmetic progression.
When determining if a sequence is arithmetic, you check whether the difference between each pair of consecutive terms remains constant. You do this by subtracting the first term from the second, the second from the third, and so forth.

For example, in a simple sequence like 2, 4, 6, 8, the differences are calculated as follows:

• 4 - 2 = 2
• 6 - 4 = 2
• 8 - 6 = 2

Here, the common difference is 2, indicating this is an arithmetic sequence.
However, if the differences vary, like in our original exercise (differences of 2, 4, 8, 16), the sequence is not arithmetic. Understanding the common difference helps determine if a sequence can be expressed in standard form.

Sequence Terms

In any sequence, each number is called a term. Sequence terms are typically represented by the symbol \( a_n \), where \( n \) signifies the position of the term in the sequence.
To find the terms in a sequence, especially arithmetic ones, a specific formula is used. For an arithmetic sequence, each term can be found using a simple addition of the previous term's common difference. However, when the sequence isn't arithmetic, like in the original exercise, you use another method specific to the pattern:

• Identify the formula: For instance, \( a_n = 4 + 2^n \)
• Substitute values: Insert integers starting from 1 to find several terms

Substituting, as seen in the original solution, showed this forms a different pattern.
Recognizing the terms and their positions helps analyze sequences effectively, especially when classifying them as arithmetic or otherwise.

Standard Form of Sequence

The standard form of an arithmetic sequence allows us to easily calculate any term by using its position. This form is expressed by the equation: \[ a_n = a + (n-1) \, d \] where:

• \( a_n \) is the \( n \)th term
• \( a \) is the first term
• \( d \) is the common difference

By substituting the values of \( a \), \( n \), and \( d \), you can find any term in the sequence without listing all preceding terms.

Unfortunately, if a sequence is not arithmetic, like the one from the original exercise where \( a_n = 4 + 2^n \) produces a series of terms with non-equal differences, this formula can't be used.
Instead, each term must be calculated using the specific rule given, without a shortcut formula like the standard arithmetic equation provides.