Problem 47 - Precalculus with Limits

Question

In Exercises 45 - 56, find the indicated \( n \)th term of the geometric sequence. 10th term: \( 5, 30, 180, \cdots \)

Step-by-Step Solution

Answer

The 10th term of the given geometric sequence is 83886080.

Step 1: Identify the Common Ratio

The common ratio in a geometric sequence is obtained by dividing any term by the previous term. Here, for example, taking the second term(metaphorically referred to as 'B') and dividing it by the first term (referred to as 'A'), we find the common ratio (r). \n So, r = B / A = 30 / 5 = 6.

Step 2: Use the Geometric Sequence Formula

The \( n \)th term of a geometric sequence can be found by using the formula: A * r^(n-1). Here, A is the first term, r is the common ratio, and n is the term number. For this problem, we need to find the 10th term, so n = 10. Substituting the known values into the formula gives: 5 * 6^(10 - 1) = 5 * 6^9.

Step 3: Do the Math

Now, compute 5 * 6^9 to get the required 10th term which is 83886080.

Key Concepts

Common Ratio

Understanding the common ratio is essential when dealing with geometric sequences. It's the multiplicative factor that relates each term to its predecessor, creating a pattern that extends throughout the sequence. In simpler terms, if we have a sequence, and we take any term after the first one and divide it by the directly preceding term, the result is what we call the common ratio, often denoted as 'r'.

For instance, if we examine the sequence in the given exercise (5, 30, 180,...), we divide the second term by the first term – that's 30 divided by 5 – giving us a common ratio of 6. This means that each term is 6 times greater than the term before it. The ability to identify the common ratio is a pivotal step in analyzing geometric sequences because it's this ratio that allows the sequence to be predictable and enables the formulation of the nth term.

Geometric Sequence Formula

The geometric sequence formula is a powerful tool that enables you to calculate any term in a geometric sequence without listing all the terms. The formula is \( a_n = a_1 \times r^{(n-1)} \). Here, \( a_n \) represents the nth term we're trying to find, \( a_1 \) is the first term of the sequence, 'r' is the common ratio, and 'n' is the term number.

Applying this formula to our exercise, where we have the first term as 5 and the common ratio as 6, to find the 10th term we plug these values along with n=10 into our formula:\[ a_{10} = 5 \times 6^{(10-1)} = 5 \times 6^9 \].Through substitution, the problem becomes a straightforward calculation, demonstrating the power and efficiency of the geometric sequence formula.

Sequence and Series

A sequence is an ordered list of numbers following a certain rule, while a series is the summation of the terms of a sequence. In the context of geometric sequences, each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. Sequences and series are fundamental concepts in mathematics, having applications across various disciplines, including finance, computer science, and physics.

In geometric sequences specifically, understanding these concepts is vital as it lays the groundwork for predicting future values and sums which are part of more complex applications such as calculating interest or modeling exponential growth. The ability to differentiate between the individual terms of a sequence and the cumulative total of a series can be crucial in practical scenarios where the sequence represents discrete events and the series represents a total outcome.