Problem 14 - Calculus: Early Transcendentals

Question

Find a formula for the general term \( a_n \) of the sequence, assuming that the pattern of the first few terms continues. $$ \left\\{\begin{array} 4, -1, \frac {1}{4}, - \frac {1}{16}, \frac {1}{64}, . . . . .\end{array}\right\\} $$

Step-by-Step Solution

Answer

The general term formula is \( a_n = (-1)^{n+1} \cdot \frac{4}{4^{n-2}} \).

Step 1: Analyze the Pattern

Observe the sequence: 4, -1, \( \frac{1}{4} \), -\( \frac{1}{16} \), \( \frac{1}{64} \), .... Notice the pattern alternates in sign and involves fractions that seem related to powers of 4.

Step 2: Determine the Denominator

Identify that the pattern for the denominators is related to powers of 4. Starting with \( 4^0 \) for the first term, \( 4^1 \) for the second (which is negative) continues with \( 4^2 \), \( 4^3 \), and \( 4^4 \), respectively.

Step 3: Determine the Sign Pattern

The sequence alternates in sign. Note that the sign changes with each term such that the 1st term is positive, the 2nd is negative, the 3rd is positive, and so forth. This can be expressed with \((-1)^{n+1}\) where n is the term number.

Step 4: Construct a General Formula

Put the components of the sequence pattern together. The general form of the term \( a_n \) can be deduced as: \[ a_n = (-1)^{n+1} \cdot \frac{1}{4^{n-2}} \] with a constant factor of 4 multiplied the alternating sign term for any positive integer \( n \geq 1 \).

Step 5: General Term Verification

Verify that the general term formula works for the initial terms:- For \( n = 1 \), \( a_1 = 4 \)- For \( n = 2 \), \( a_2 = -1 \)- For \( n = 3 \), \( a_3 = \frac{1}{4} \)All calculations confirm the pattern, verifying the constructed formula.

Key Concepts

Alternating Signs

In some numerical sequences, you might notice a pattern where the signs of the terms alternate between positive and negative. This is crucial for understanding the structure of a sequence. In our given problem, the sequence starts with a positive sign and each subsequent term flips its sign:

• The first term is positive.
• The second term is negative.
• The third term goes back to positive, and so forth.

This alternating sign can be expressed using a mathematical expression involving the exponent of defined as \((-1)^{n+1}\). This formula becomes very handy as it captures the pattern of alternating signs perfectly. For example, when \( n \) is odd, the term is positive, and when \( n \) is even, the term is negative. Understanding how this formula works helps in finding the general term for such types of sequences.

Powers of 4

The sequence in our problem involves fractions related to powers of 4. This concept forms the foundation of the sequence's structure. Initially, each term's denominator appears to be a power of 4:

• First term: no denominator involved.
• Second term: denominator is \(4^1\).
• Third term: denominator is \(4^2\), and so on.

Recognizing this power pattern helps unlock how each denominator is derived. For any term \(n\), the denominator can be expressed as\(4^{n-2}\). This transformation simplifies the process of constructing the general term and highlights the constant growth of the denominator as the term number increases. Thus, embracing this power pattern is key to understanding the sequence and predicting future terms.

General Term

Finding the general term of a sequence allows us to compactly represent and calculate any term in the sequence without listing all previous terms. In this sequence, the general term \(a_n\) encompasses both the alternating sign component and the power of 4 component. By combining both components identified:

• The alternating sign is given by \((-1)^{n+1}\).
• The power of 4 appears in the denominator as \(4^{n-2}\).

We can assemble these elements into the comprehensive form:\[a_n = (-1)^{n+1} \cdot \frac{1}{4^{n-2}}\]This formula efficiently calculates any term's value, encapsulating the sequence’s behavior. Moreover, it includes a constant factor, in this sequence's case a multiplication with 4, to match the sequence's growth pattern fully. Through systematic examination, we ensure this general term adapts to any positive integer \( n \), minimizing errors in predicting succeeding terms.

Mathematical Pattern

Recognizing mathematical patterns is an essential skill in understanding sequences and series. The given sequence features both alternating signs and denominators that correspond to powers of 4, forming a recognizable pattern. By identifying these interconnected elements:

• Sign alternation - Formulaic representation through integer powers of -1.
• Powers of 4 - Denominators developed as powers increasing with each term.

These features supplement each other to create richer mathematical insights. Breaking down the sequence into these distinct patterns allows us to formulate the accurate general term, predictive of further terms. Spotting these patterns not only simplifies mathematical problems but also enhances our capacity to solve complex exercises efficiently. Mastering such observations leads to effective problem-solving approaches, turning seemingly complicated sequences into manageable expressions.