Problem 38 - Algebra for College Students

Question

Write a formula for the general term of each infinite sequence. \(1,-3,9,-27, \dots\)

Step-by-Step Solution

Answer

The general term is given by: a_n = (-3)^{n-1}

Step 1: Identify the pattern

Observe the given sequence: 1, -3, 9, -27, ... Note that each term alternates in sign and appears to be a multiple of the previous term by -3.

Step 2: Express terms as powers

Rewrite the terms in terms of powers of -3: 1 = (-3)^0, -3 = (-3)^1, 9 = (-3)^2, -27 = (-3)^3, ...

Step 3: Set up the general term

Based on the pattern identified, the general term of the sequence can be written as: a_n = (-3)^{n-1}Here, 'n' is the position of the term in the sequence.

Step 4: Verify the formula

Check if the formula works for a few terms: For n=1: a_1 = (-3)^{1-1} = 1For n=2: a_2 = (-3)^{2-1} = -3For n=3: a_3 = (-3)^{3-1} = 9The formula correctly produces the terms 1, -3, and 9.

Key Concepts

Infinite Sequences

An infinite sequence is a list of numbers that continues forever. Unlike finite sequences, which have a fixed number of terms, infinite sequences go on endlessly. Here, our sequence is 1, -3, 9, -27, ... This sequence does not terminate and will keep adding more terms as you go further.

Understanding infinite sequences is crucial because they offer a way to describe patterns and behaviors in mathematics that do not have an endpoint. You might encounter such sequences in various mathematical contexts, including series, summations, and more advanced topics.

Pattern Identification

One of the first steps in tackling sequences is identifying the pattern. This involves looking closely at the terms given in the sequence.

Here, we observe the sequence: 1, -3, 9, -27, ... At first glance, you can see that the terms alternate in sign (positive, negative, positive, negative). Recognizing patterns like alternating signs or common ratios helps us find the formula that generates the terms.

In our case, each term is a multiple of the previous one by -3. This consistent factor indicates a clear, predictable pattern.

Powers of Numbers

To express the terms clearly, it can be helpful to use powers of numbers. In the sequence 1, -3, 9, -27, ..., one way to describe each term is through the power of a base number.

By rewriting the terms as powers of -3, we get:

• 1 as \( (-3)^0 \)
• -3 as \( (-3)^1 \)
• 9 as \( (-3)^2 \)
• -27 as \( (-3)^3 \)

This representation makes it easier to see that the term in position n is \(( -3 )^{n-1} \).

Alternating Signs

Alternating signs in sequences imply that the terms switch between positive and negative. This pattern is evident in sequences like 1, -3, 9, -27, ...

Detecting alternating signs is valuable when writing a formula. In our sequence, the sign changes consistently with each term. Using a base raised to a power, like \(( -3 )^{n-1} \), naturally incorporates the alternating sign because raising -3 to an odd power yields a negative term, while an even power yields a positive term.

By understanding these patterns, you can write a formula that accurately captures the essence of the sequence, ensuring that each term is correctly placed in the sequence with the proper sign.