Problem 26 - Precalculus: Mathematics for Calculus

Question

Find the first five terms of the sequence, and determine whether it is geometric. If it is geometric, find the common ratio, and express the \(n\) th term of the sequence in the standard form \(a_{n}=a r^{n-1}.\) $$a_{n}=(-1)^{n} 2^{n}$$

Step-by-Step Solution

Answer

The sequence is geometric with common ratio -2; the nth term is \(a_n = -2(-2)^{n-1}\).

Step 1: Find the First Five Terms

To find the first five terms of the sequence given by \( a_n = (-1)^n 2^n \), substitute \( n = 1, 2, 3, 4, \) and \( 5 \) into the formula:- For \( n = 1 \), \( a_1 = (-1)^1 2^1 = -2 \).- For \( n = 2 \), \( a_2 = (-1)^2 2^2 = 4 \).- For \( n = 3 \), \( a_3 = (-1)^3 2^3 = -8 \).- For \( n = 4 \), \( a_4 = (-1)^4 2^4 = 16 \).- For \( n = 5 \), \( a_5 = (-1)^5 2^5 = -32 \).Thus, the first five terms of the sequence are \(-2, 4, -8, 16, -32\).

Step 2: Check for Geometric Sequence

To check if the sequence is geometric, find the ratio of the consecutive terms. - From \(a_2\) to \(a_1\), the ratio is \( \frac{4}{-2} = -2 \).- From \(a_3\) to \(a_2\), the ratio is \( \frac{-8}{4} = -2 \).- From \(a_4\) to \(a_3\), the ratio is \( \frac{16}{-8} = -2 \).- From \(a_5\) to \(a_4\), the ratio is \( \frac{-32}{16} = -2 \).Since the ratio \(-2\) remains constant, the sequence is geometric with a common ratio of \(-2\).

Step 3: Express the nth Term in Standard Form

Since the sequence is geometric with common ratio \( r = -2 \) and first term \( a_1 = -2 \), the general form of the nth term can be expressed as:\[ a_n = a_1 r^{n-1} \]Substitute the values:\[ a_n = -2 (-2)^{n-1} \]

Key Concepts

Common Ratio

A common ratio is a key feature of a geometric sequence. It is the constant factor between each pair of consecutive terms in the sequence. To identify if a series is geometric, we examine this focal point.
Let’s look at the given sequence:

• The first five terms are \, \(-2, 4, -8, 16, -32\).
• To find the common ratio, divide each term by the previous one.
• For example: the ratio from the second term to the first term is \( \frac{4}{-2} = -2 \).
• Check each pair: \( \frac{-8}{4} = -2 \), \( \frac{16}{-8} = -2 \), and \( \frac{-32}{16} = -2 \).

Clearly, the common ratio for this sequence is \(-2\).
This constant factor confirms that the sequence is geometric.
In a geometric sequence, maintaining the same ratio across terms is essential. It ensures that each term after the first is derived from multiplying the previous term by the common ratio.

Sequence Terms

The sequence terms represent the individual elements of a sequence. In our case, they are derived from the expression \(a_{n} = (-1)^{n} 2^{n}\).

• Each term of the sequence is generated by substituting successive values of \(n\) into the expression.
• For \(n = 1\), the term is \(-2\). For \(n = 2\), it's \(4\), followed by \(-8, 16, -32\) for \(n = 3, 4, 5\), respectively.
• This pattern of alternating positive and negative numbers arises from the \((-1)^{n}\) part of the formula.

Sequence terms form the building blocks of a sequence.
Understanding their calculation is vital to grasp the overall pattern and nature of the sequence. The formula must be followed carefully for each term to ensure precision.
Comprehending these terms enables the analysis of their distribution and behavior over time.

Geometric Progression

A geometric progression is a type of sequence where each term is found by multiplying the previous term by a specific non-zero constant. This constant is known as the common ratio.
In our example:

• We confirmed that \, \(-2, 4, -8, 16, -32\) is a geometric sequence.
• The consistent multiplication by the common ratio of \(-2\) indicates this progression.

Geometric sequences can be expressed using a general formula for the \(n\)th term.
The formula: \[ a_n = a_1 r^{n-1} \]This allows for easy calculations for consecutive terms.
Using the first term \(a_1 = -2\) and the common ratio \(r = -2\), the nth term is given by:\[ a_n = -2 (-2)^{n-1} \]
This standard form simplifies predicting terms farther along the sequence. It demonstrates the layered multiplication effect of the common ratio. Geometric progression is powerful in various mathematical and real-life applications, showcasing repetitive, scaling patterns.