Problem 14 - Contemporary Precalculus

Question

In Exercises \(13-22,\) one term and the common ratio r of a geometric sequence are given. Find the sixth term and a formula for the nth term. $$a_{1}=1, r=-2$$

Step-by-Step Solution

Answer

Answer: The sixth term of the geometric sequence is -32 and the formula for the nth term is \(a_n = 1 \cdot (-2)^{n-1}\).

Step 1: Identify the given information

We are given the following information: - First term, \(a_1 = 1\) - Common ratio, \(r = -2\)

Step 2: Use the formula to find the sixth term, \(a_6\)

Using the formula for the nth term of a geometric sequence, we can find the sixth term: $$a_6 = a_1 \cdot r^{6-1}$$ Plugging in the given values, we have: $$a_6 = 1 \cdot (-2)^{5}$$ Calculating the value, we get: $$a_6 = 1 \cdot (-32) = -32$$ So, the sixth term, \(a_6\), is -32.

Step 3: Write down the formula for the nth term, \(a_n\)

Now that we have the values of \(a_1\) and \(r\), we can write the formula for the nth term of the geometric sequence: $$a_n = a_1 \cdot r^{n-1}$$ Plugging in the given values, we have: $$a_n = 1 \cdot (-2)^{n-1}$$ This is the formula for the nth term of the given geometric sequence.

Key Concepts

Understanding the nth Term Formula for Geometric Sequences

The nth term formula is a powerful tool in the study of geometric sequences. A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the "common ratio." This formula helps us easily find any specified term in the sequence without listing all the previous terms.
This is especially useful for large sequences and saves time and effort.
The formula for the nth term of a geometric sequence is:

• \(a_n = a_1 \cdot r^{n-1}\)

Here:

• \(a_n\) is the nth term we want to find.
• \(a_1\) is the first term of the sequence.
• \(r\) is the common ratio.
• \(n\) is the number of the term we are interested in.

In our given exercise, we started with \(a_1 = 1\) and a common ratio of \(r = -2\). To find the sixth term \(a_6\), we used the formula:

• \(a_6 = 1 \cdot (-2)^{5} = -32\)

This demonstrates the simplicity of substituting our known values into the nth term formula to calculate any term in the sequence.

Exploring Geometric Series

A geometric series is the sum of the terms in a geometric sequence. Understanding this concept provides insights into patterns and structures that can emerge from repeated multiplication of the common ratio.
In contrast to just knowing individual terms, a series gives us a single number that can characterize the entire collection of terms.
The sum of the first \(n\) terms of a geometric sequence, also known as a geometric series, can be found using the formula:

• \(S_n = a_1 \frac{1-r^n}{1-r}\) , when \(r eq 1\)

This formula allows us to determine the total of the sequence up to any point. Understanding how each term contributes to the total requires insight into the nature of multiplication and progression in sequences. In the given exercise, while we focused on terms of the sequence, knowing the concept of a series is also essential for complete mastery over sequences, especially for advanced mathematical contexts like finance or physics.

Decoding the Common Ratio

The common ratio is a fundamental part of understanding geometric sequences. It is the constant factor between consecutive terms in a sequence. Knowing the common ratio allows us to predict the behavior of the entire sequence once the first term is known.
It acts as the bridge between each term and defines the exponential nature of the sequence.
In our provided exercise, the common ratio \(r = -2\) dictates how each term is derived from the previous one. Starting from the first term \(a_1 = 1\):

• The second term is \(1 \cdot (-2) = -2\)
• The third term is \(-2 \cdot (-2) = 4\)
• And this pattern continues as \(4 \cdot (-2) = -8\), \(-8 \cdot (-2) = 16\), etc.

This multiplicative factor of \(-2\) shows a sequence that alternates between positive and negative values and grows in absolute magnitude exponentially. Understanding the role of the common ratio is key to grasping how quickly or slowly a sequence will grow or shrink and what patterns it will follow across its terms.