Problem 28 - Precalculus: Mathematics for Calculus

Question

Find the first five terms of the sequence, and determine whether it is arithmetic. If it 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, so we can't determine a common difference or use the form \(a_n = a + (n-1)d\).

Step 1: Calculate the First Five Terms

To find the first five terms, we substitute \(n = 1, 2, 3, 4,\) and \(5\) into the formula \( a_n = 4 + 2^n \). Thus, we have:- 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, and 36.

Step 2: Check if the Sequence is Arithmetic

To determine if the sequence is arithmetic, check if the difference between consecutive terms is constant. Calculate the differences:- 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 \( a_4 \): \( 36 - 20 = 16 \)Since these differences are not equal, the sequence is not arithmetic.

Step 3: Conclude on Sequence Type and Representation

Since the sequence is not arithmetic (the differences between consecutive terms are not constant), we cannot express the \(n\)th term of the sequence in the form \(a_n = a + (n-1)d\). The sequence \(a_n = 4 + 2^n\) does not fit the arithmetic sequence criteria but fits an exponential growth pattern due to the \(2^n\) component.

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a series of numbers in which the difference between any two consecutive terms remains constant. This constant difference is key to identifying an arithmetic sequence. If this difference, known as the "common difference," is present throughout the series, then the sequence is deemed arithmetic.
Understanding whether a sequence is arithmetic helps in predicting future terms using a specific formula that is straightforward and logical. While arithmetic sequences offer simplicity, they differ vastly from other sequence types due to this consistent pattern.
Arithmetic sequences are integral in various mathematical applications, especially when dealing with linear progressions and straightforward computations.

Common Difference

The common difference is the fixed amount added to or subtracted from each term in an arithmetic sequence to get the next term. This distinctive feature forms the backbone of an arithmetic sequence.

• If the difference remains the same, it confirms that the sequence is arithmetic.
• For instance, in a sequence like 2, 4, 6, 8, each term is obtained by adding 2, so the common difference is 2.

Finding the common difference involves subtracting any term from the subsequent term. If the result is consistent across the sequence, it reaffirms the arithmetic nature.
In sequences that aren't arithmetic, the difference will vary, indicating a different pattern or no consistent pattern at all.

Exponential Growth

Exponential growth indicates a sequence where each term increases by a multiplication factor, rather than by a constant addition. This growth pattern signifies frequent doubling or rapid increase, resembling real-world exponential phenomena like population growth or compound interest.
In our exercise, the term formula includes a base component added to an exponential term, specifically the part involving powers of 2.

• The term formula: \( a_n = 4 + 2^n \), highlights this by multiplying 2 by itself for each increment in \( n \).

Such sequences increase swiftly as \( n \) grows larger.
Understanding exponential growth in mathematics provides insight into how quickly things can escalate in various situations, emphasizing non-linear progression over linear patterns.

Nth Term Formula

The nth term formula provides a method to calculate any term in a sequence without listing all the previous terms. For arithmetic sequences, the formula is typically \( a_n = a + (n-1) \, d \), where:

• \( a \) is the first term of the sequence.
• \( d \) is the common difference.
• \( n \) is the term number.

However, when we encounter non-arithmetic sequences, such as those exhibiting exponential growth, the formula aligns differently. In these instances, the sequence cannot be expressed in simple additive terms but rather involves more complex formulations, like \( a_n = 4 + 2^n \).
This approach allows for flexibility across different sequence types and aids in precise calculation, crucial for both theoretical and practical scenarios.