Problem 22 - Algebra and Trigonometry

Question

21-26 \(\approx\) 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.

Step 1: Determine the First Five Terms

The rule for the nth term is given as \( a_n = 4 + 2^n \). To find the first five terms, we simply substitute \( n = 1, 2, 3, 4, \) and \( 5 \):\[a_1 = 4 + 2^1 = 4 + 2 = 6\]\[a_2 = 4 + 2^2 = 4 + 4 = 8\]\[a_3 = 4 + 2^3 = 4 + 8 = 12\]\[a_4 = 4 + 2^4 = 4 + 16 = 20\]\[a_5 = 4 + 2^5 = 4 + 32 = 36\]The first five terms of the sequence are 6, 8, 12, 20, and 36.

Step 2: Check if the Sequence is Arithmetic

An arithmetic sequence has a constant difference \(d\) between consecutive terms. Calculate the differences between the consecutive terms:\[8 - 6 = 2 \12 - 8 = 4 \20 - 12 = 8 \36 - 20 = 16\]The differences are not constant (2, 4, 8, 16), so this sequence is not arithmetic.

Step 3: Conclude Non-Arithmetic Status

Since the differences between consecutive terms are not constant, the sequence is not arithmetic. Therefore, we do not have a common difference or a standard form for an arithmetic progression.

Key Concepts

Understanding Arithmetic Sequences

An arithmetic sequence is a sequence of numbers with a clear pattern. In such a sequence, each term is derived by adding or subtracting a constant value to the previous term. This constant is known as the 'common difference'. Because of this predictable pattern, arithmetic sequences are commonly introduced early in mathematics courses to help students understand the idea of sequences thoroughly.
The basic rule of an arithmetic sequence holds that every term after the first is the sum of the previous term and the common difference. This consistency allows easy predictions and calculations for missing terms. In simpler terms, if you know the common difference and the first term, you can generate any number in the sequence simply by repeatedly adding the common difference.

• Example: A simple arithmetic sequence could be 5, 8, 11, 14, 17, where each term is 3 more than the previous term.

When confronted with any sequence problem, checking if the sequence is arithmetic is usually one of the first steps. This process involves identifying if there exists a constant difference between consecutive terms.

Explaining Common Difference

The 'common difference' is one of the core concepts that define an arithmetic sequence. It is the difference between any two successive terms. This difference remains constant throughout the sequence, making it an essential element in determining and defining arithmetic sequences.

• For instance, in the sequence 3, 6, 9, 12, the common difference is 3.
• If the difference between terms in a sequence is not consistent, like in the examples shared above with differences of 2, 4, 8, and 16, the sequence is not considered arithmetic.

The consistency of the common difference enables the predictable nature of arithmetic sequences. You can always rely on this constant to find the subsequent terms. Thus, identifying a common difference is crucial when working with sequences to understand the structure and predictability of arithmetic progressions.
Unfortunately, the sequence given in the exercise does not have a common difference, indicating it is not arithmetic.

Determining the nth Term of a Sequence

One of the exciting parts of working with sequences is the ability to determine any term in the sequence, often referred to as the "nth term." For arithmetic sequences, the formula is relatively straightforward: \[ a_n = a_1 + (n-1) \cdot d \]where \(a_n\) represents the nth term, \(a_1\) is the first term, \(n\) is the term number, and \(d\) is the common difference.
By using this formula, you can quickly calculate any term in a sequence without having to list all previous terms. It saves time and effort, especially for sequences with a large number of terms.
However, this formula specifically applies to arithmetic sequences where the common difference remains the same between terms. Since the given sequence in this exercise does not exhibit a constant difference, this formula is not applicable. As a consequence, predicting terms in non-arithmetic sequences can become more complex, often requiring more detailed calculations or even different formulas tailored to suit non-linear patterns.