Problem 26 - Algebra and Trigonometry

Question

\(25-32\) . Find the \(n\) th 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 \(a_n = (-1)^{n+1} \cdot \frac{1}{3^n}\).

Step 1: Identify the Pattern

Look at the sequence: \(-\frac{1}{3}, \frac{1}{9}, -\frac{1}{27}, \frac{1}{81}, \dots\). Notice that the denominators are powers of 3: \(3^1, 3^2, 3^3, 3^4, \dots\). This suggests a pattern in the powers of the denominator.

Step 2: Determine the Sign Pattern

The sequence terms alternate in sign: negative, positive, negative, positive, and so on. This alternation can be represented by a factor of \((-1)^n\) where \(n\) is the term number. Specifically for this sequence, we use \((-1)^{n+1}\) because the first term is negative.

Step 3: Establish the General Formula

Combine the alternating sign factor and the denominator pattern to write the general formula. The denominators in the sequence align with \(3^n\). Hence, the sequence can be expressed as: \(a_n = (-1)^{n+1} \times \frac{1}{3^n}\).

Step 4: Write the Formula for the nth Term

According to our analysis: \[ a_n = (-1)^{n+1} \cdot \frac{1}{3^n} \]This formula gives the nth term of the sequence, taking into account both the alternating signs and the pattern in the denominators.

Key Concepts

Alternating Signs

In the given sequence \(-\frac{1}{3}, \frac{1}{9}, -\frac{1}{27}, \frac{1}{81}, \ldots\), the terms alternate between negative and positive values. Alternating signs are a common feature in sequences, creating a zigzag pattern of sorts. To incorporate alternating signs mathematically, we use terms like \[(-1)^{n+1}\]. This expression ensures the sequence changes sign with each increment in \(n\).

• The power \(n+1\) keeps the initial term negative.
• When \(n\) is even, \((-1)^{n+1} = -1\), yielding negative terms.
• When \(n\) is odd, \((-1)^{n+1} = 1\), producing positive terms.

This construction is vital in our sequence to maintain the sign alternation pattern reflected in each term.

Powers of a Number

In sequences, numbers often follow a pattern of exponential growth or reduction, usually indicated by powers of a base number. In this sequence, denominators exhibit powers of 3: \(3^1, 3^2, 3^3, 3^4,\) and so on. Recognizing these powers:

• \(3^1 = 3\)
• \(3^2 = 9\)
• \(3^3 = 27\)
• \(3^4 = 81\)

The pattern of these powers indicates each term's denominator forms a sequence based on the base number 3. In general, such sequences utilize the formula \(b^n\), where \(b\) is the base (3 in this case), and \(n\) is the position of the term in the sequence. Understanding which component grows (or shrinks) exponentially is essential for finding the term's complete expression.

General Term Formula

Once a pattern is identified in a sequence, writing a general formula to encapsulate all terms is crucial. For this sequence, with alternating signs and powers of 3, we need a combination of these behaviors:The nth term, \(a_n\), can be given by the formula \[a_n = (-1)^{n+1} \cdot \frac{1}{3^n}\].

• The factor \((-1)^{n+1}\) ensures the alternation of signs, as discussed earlier.
• The expression \(\frac{1}{3^n}\) represents the power pattern of the denominators.

This formula is a compact way to determine any term in the sequence based on its position, effectively combining all observed patterns. A well-constructed formula reveals the mathematics behind seemingly complex sequences, empowering you to predict and analyze them with clarity.