Problem 26 - Precalculus Mathematics for Calculus

Question

Find the nth term of a sequence whose first several terms are given. $$-\frac{1}{3}, \frac{1}{9},-\frac{1}{27}, \frac{1}{81}, \dots$$

Step-by-Step Solution

Answer

The nth term is \(a_n = \frac{(-1)^n}{3^n}\).

Step 1: Identify the Pattern in the Sequence

The given sequence is \(-\frac{1}{3}, \frac{1}{9}, -\frac{1}{27}, \frac{1}{81}, \dots\). Notice the numerators are alternating signs and the denominators are increasing powers of 3: \(3^1, 3^2, 3^3, 3^4, \dots\).

Step 2: Determine the Common Ratio

This sequence appears to follow a geometric pattern. To find the common ratio \(r\), divide any term by the previous term. For example, \(r = \frac{1/9}{-1/3} = -\frac{1}{3}\).

Step 3: Write the General Formula for a Geometric Sequence

A geometric sequence is given by the formula: \(a_n = a_1 \times r^{n-1}\), where \(a_1\) is the first term and \(r\) is the common ratio. In this sequence, \(a_1 = -\frac{1}{3}\) and \(r = -\frac{1}{3}\).

Step 4: Plug Values into the Formula

Substitute \(a_1 = -\frac{1}{3}\) and \(r = -\frac{1}{3}\) into the general formula: \(a_n = -\frac{1}{3} \times (-\frac{1}{3})^{n-1}\).

Step 5: Simplify the Formula

Simplify the expression further: \(a_n = (-1) \times \frac{1}{3^n} = \frac{(-1)^n}{3^n}\). This formula represents the \(n\)th term of the sequence.

Key Concepts

Common Ratio

In a geometric sequence, the common ratio is crucial, as it determines the multiplicative factor between successive terms. For a sequence to be geometric, the ratio between any term and the previous one must be constant. This characteristic ratio is called the 'common ratio'. To find it, simply divide one term by the preceding term.

This property allows for quick identification and concept understanding. If you have a sequence \(-\frac{1}{3}, \frac{1}{9}, -\frac{1}{27}, \frac{1}{81}, \dots\), just choose any two consecutive terms. For example, dividing \(\frac{1}{9}\) by \(-\frac{1}{3}\) gives a common ratio of \(-\frac{1}{3}\).

Recognizing this common ratio as consistent across the sequence confirms its geometric nature. This understanding brings us to the next vital component: defining elements of sequence using mathematical formulas.

Nth Term Formula

The nth term formula is our key tool for defining any term within a geometric sequence without list enumeration. For any geometric sequence, the nth term, denoted as \(a_n\), is defined by the formula:

• \(a_n = a_1 \times r^{n-1}\)

Here, \(a_1\) is the first term and \(r\) is the common ratio.

This formula makes it easy to calculate any term in the sequence. For the sequence \(-\frac{1}{3}, \frac{1}{9}, -\frac{1}{27}\), suppose you want the fifth term. With \(a_1 = -\frac{1}{3}\) and \(r = -\frac{1}{3}\), plug these values into the formula to get \(a_n = -\frac{1}{3} \times (-\frac{1}{3})^{n-1}\).

This flexibility simplifies work with far terms without manually listing every step in between.

Pattern Recognition

Recognizing patterns in sequences plays a crucial role in sequence analysis. Essentially, this involves observing regularities in order behavior, such as alternating signs or consistent ratios. Let's revisit our example sequence: \(-\frac{1}{3}, \frac{1}{9}, -\frac{1}{27}, \frac{1}{81}, \dots\).

Observe how it alternates between negative and positive values due to the term's numerator. The denominators consistently increase as powers of 3: \(3^1, 3^2, 3^3, 3^4\), and so on. Recognizing this gradual pattern is key to correctly identifying both the sequence type and deriving its nth term formula.

Such insight helps decode the complex sequences quickly and efficiently. Anyone trying to solve similar problems can develop intuitive reasoning for these key patterns for not just this exercise but future challenges as well.

Sequence Analysis

Sequence analysis includes breaking down a sequence into identifiable components and determining its underlying structure. It covers multiple aspects, such as its common ratio and specifics of its term formations. In the given sequence, by understanding both the consistent ratio and sign alternation, we can discern critical features of the series.

This sequence's analysis would involve:

• Determining each term's formula like \(\frac{(-1)^n}{3^n}\).
• Observing how Alternating signs follow pattern \( (-1)^n \).
• The consistency in denominator growth, indicating geometric nature.

By dissecting sequences into their base patterns and terms, sequence analysis empowers learners with potential for deeper insights. It transforms simple recognition into a full analysis, paving the way for both improved understanding and academic success.