Problem 18 - College Algebra

Question

Write a formula for the general term (the nth term) of each geometric sequence. Then use the formula for \(a_{n}\)to find \(a_{7},\) the seventh term of the sequence. $$3,15,75,375, \dots$$

Step-by-Step Solution

Answer

The 7th term \(a_7\) in the sequence is \(3 * 5^6 = 9375\)

Step 1: Identify the common ratio and the first term

In our sequence 3, 15, 75, 375, we see that each term is multiplied by 5 to reach the next term. Thus, our common ratio \(r\) is 5 and the first term \(a_1\) is 3.

Step 2: Apply the formula

Now that we know the common ratio and the first term, we can apply the formula \(a_n = a_1 * r^{(n-1)}\) to find the nth term.

Step 3: Calculate the 7th term

For \(a_7\) , the formula becomes \(a_7 = 3 * 5^{(7-1)}\). Then we calculate \(5^{6}\) and multiply by 3 to get \(a_7\) term.

Key Concepts

Geometric Sequence Formula

Understanding the geometric sequence formula is a critical step in solving problems related to geometric progressions. A geometric sequence is a series of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

For instance, given the sequence in the exercise, the common ratio is consistent. The formula to find any term in the sequence, known as the nth term, is given by:
gn = a1 * r^(n-1).
Here, a1 is the first term, r is the common ratio, and n denotes the term number. Using this formula simplifies the process of finding any term in the sequence, preventing the need for manual, multi-step calculations that would occur if we were to multiply the common ratio by each preceding term successively.

Common Ratio

The common ratio is the backbone of a geometric sequence, representing the factor by which we multiply each term to achieve the next one in the sequence. The common ratio is constant throughout the sequence and is a crucial element in determining its nature and behavior.

In our example, to find the common ratio, observe the relationship between consecutive terms. Dividing any term by its preceding term in the sequence 15 / 3 = 5, 75 / 15 = 5, confirms that the common ratio is 5. This value is then utilized as 'r' in our geometric sequence formula to find subsequent terms of the sequence. The common ratio can be positive or negative, which affects whether the sequence increases or decreases, and if it alternates sign.

Nth Term of a Sequence

Determining the nth term of a geometric sequence is an essential skill for many mathematical and real-world applications. The nth term can be thought of as a function that takes the term number n as input and returns the value of that term as output.

As we have established, using the formula gn = a1 * r^(n-1), we can find any term in the sequence without listing all terms, which becomes incredibly efficient for large values of n. In our practice exercise, we calculated the 7th term (a7) by substituting n = 7, a1 = 3, and r = 5 into the formula. This process allowed us to quickly determine the 7th term without manually multiplying by the common ratio six times. The ability to find an directly is particularly valuable in predicting future terms or investigating the properties of the sequence at a large scale.