Problem 13 - 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. $$a_{1}=5, r=2$$

Step-by-Step Solution

Answer

Answer: The sixth term in the geometric progression is 160 and the formula for the nth term is \(a_n = 5 * 2^{n-1}\).

Step 1: Identify the given values

We are given the first term (\(a_1\)) and the common ratio (r): \(a_1 = 5\) \(r = 2\)

Step 2: Find the sixth term

To find the sixth term (\(a_6\)), we will use the formula \(a_n = a_1 * r^{n-1}\) with \(n=6\). \(a_6 = a_1 * r^{6-1} = 5 * 2^{5} = 5 * 32 = 160\) So the sixth term is 160.

Step 3: Find the formula for the nth term

We already have the formula for the nth term of a geometric progression, which is: \(a_n = a_1 * r^{n-1}\) Since we know both the first term, \(a_1 = 5\), and the common ratio, \(r=2\), we can express the formula for the nth term as follows: \(a_n = 5 * 2^{n-1}\) So the formula for the nth term is \(a_n = 5 * 2^{n-1}\).

Key Concepts

Common Ratio

The concept of a 'common ratio' is fundamental when we dive into the world of geometric sequences. Think of a geometric sequence as a chain of numbers where each link is produced by multiplying the previous one by a fixed, non-zero number known as the common ratio, denoted by 'r'. In simple terms, if you start with a number and keep multiplying it by the common ratio, you'll get a geometric sequence.

For instance, if you begin with 5 and repeatedly multiply by 2 (the common ratio in our exercise), you’ll get the sequence 5, 10, 20, 40, 80, ... and so on. This multiplication factor determines how quickly the sequence grows or shrinks, which is why identifying the common ratio is key to understanding the behavior of the sequence.

Nth Term Formula

The 'nth term formula' is a powerful tool for pinpointing any term in a geometric sequence without having to list out all the preceding terms. It empowers students to leapfrog directly to any term they wish to find. Mathematically, the nth term ('a_n') of a geometric sequence can be calculated using the formula: \(a_n = a_1 * r^{n-1}\).In this formula, 'a_1' represents the first term of the sequence, 'r' is the common ratio, and 'n' is the term number you want to find. To bring this to life, let's say you wanted to find the 6th term (as in your exercise). Plugging the values into the formula gives you 5 (the first term) multiplied by 2 (the common ratio) raised to the power of 5 (which is 6 minus 1), leading you to the 6th term, 160. Understanding this nth term formula means you essentially have a quick shortcut to any term in the sequence.

Geometric Progression

A 'geometric progression' is essentially a fancy term for a geometric sequence. It is a series of numbers where each term after the first is found by multiplying the previous term by the common ratio. These types of sequences show up frequently in various domains—like finance, computing, and physics—often describing scenarios with exponential growth or decay.

Understanding a geometric progression is not just about knowing the terms; it involves grasping how they escalate or diminish. It embodies the power of exponential change, which is much different from the steady increase seen in arithmetic sequences. The progression provides a clear picture of how each term is connected and interdependent, emphasizing the significance of initial values and proportionality represented by the common ratio.