Problem 18 - Precalculus: Mathematical for Calculus

Question

Find the first five terms of the sequence and determine if 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.

Step 1: Identify the Sequence Formula

The sequence is given by the formula \( a_n = 4 + 2^n \). This is a general term to find each term in the sequence.

Step 2: Calculate the First Term

Substitute \( n = 1 \) into the formula: \( a_1 = 4 + 2^1 = 4 + 2 = 6 \). Thus, the first term is 6.

Step 3: Calculate the Second Term

Substitute \( n = 2 \) into the formula: \( a_2 = 4 + 2^2 = 4 + 4 = 8 \). Thus, the second term is 8.

Step 4: Calculate the Third Term

Substitute \( n = 3 \) into the formula: \( a_3 = 4 + 2^3 = 4 + 8 = 12 \). Thus, the third term is 12.

Step 5: Calculate the Fourth Term

Substitute \( n = 4 \) into the formula: \( a_4 = 4 + 2^4 = 4 + 16 = 20 \). Thus, the fourth term is 20.

Step 6: Calculate the Fifth Term

Substitute \( n = 5 \) into the formula: \( a_5 = 4 + 2^5 = 4 + 32 = 36 \). Thus, the fifth term is 36.

Step 7: List the First Five Terms

The first five terms of the sequence are: 6, 8, 12, 20, 36.

Step 8: Check for Arithmetic Sequence

To determine if the sequence is arithmetic, find the differences between consecutive terms: \( 8 - 6 = 2 \), \( 12 - 8 = 4 \), \( 20 - 12 = 8 \), \( 36 - 20 = 16 \). Since the differences are not constant, the sequence is not arithmetic.

Key Concepts

Arithmetic Sequences

An arithmetic sequence is a type of sequence where each term after the first is obtained by adding a fixed constant called the common difference to the previous term. Here are some key characteristics:

• The formula for the general term of an arithmetic sequence is: \( a_n = a_1 + (n-1) \cdot d \), where \( a_1 \) is the first term and \( d \) is the common difference.
• If the difference between consecutive terms is constant, the sequence is classified as arithmetic.
• Examples: The sequence 2, 5, 8, 11 is arithmetic with a common difference of 3.

In arithmetic sequences, predicting future terms becomes straightforward by using the formula repeatedly. When you come across any sequence, always check the difference between terms to establish whether it is arithmetic or not.

Sequence Formula

The sequence formula is a mathematical expression used to generate the terms of a sequence. It enables us to compute any term in the sequence without listing all preceding terms. Here’s what you need to know:

• It is an expression that involves a mathematical function or operations depending on the nature of the sequence, such as linear (arithmetic) or exponential.
• For arithmetic sequences, it's \( a_n = a_1 + (n-1) \cdot d \), allowing you to find any term directly.
• Understanding the sequence formula helps in finding patterns, simplifying calculations, and identifying whether a sequence is arithmetic or non-arithmetic.

In the given exercise, the sequence formula \( a_n = 4 + 2^n \) indicates an exponential change, affecting how each term expands.

Non-arithmetic Sequences

Non-arithmetic sequences do not have a constant difference between consecutive terms. This means they cannot be generated by repeatedly adding the same number to the previous term. Here are some of their features:

• They can grow exponentially like the sequence \( a_n = 4 + 2^n \), indicating each term is the result of the previous term multiplied by a variable factor.
• Examples include geometric sequences and other sequences described by non-linear formulas.
• With non-arithmetic sequences, discerning a pattern might be harder compared to arithmetic ones since they don't have a straightforward additive pattern.

In practice, recognizing a non-arithmetic sequence is essential when dealing with complex patterns that require more than simple addition for term calculation.