Problem 56 - Precalculus with Limits

Question

In Exercises 45 - 56, find the indicated \( n \)th term of the geometric sequence. 9th term: \( a_3 = 11, a_4 = -11\sqrt{11} \)

Step-by-Step Solution

Answer

The 9th term of the geometric sequence is \( 11^{4} \).

Step 1: Finding the common ratio

The common ratio \( r \) can be found by dividing consecutive terms. Thus, \( r = a_4/a_3 = -11\sqrt{11}/11 = -\sqrt{11}\).

Step 2: Finding the first term

Rewrite the expression for \( a_n \) as \( a_1 = a_n / (r^{(n-1)}) \). Plug in 3 for \( n \), 11 for \( a_3 \), and \( -\sqrt{11} \) for \( r \) then solve: \( a_1 = 11 / ((-\sqrt{11})^{(3-1)}) = \sqrt{11}\).

Step 3: Calculating the 9th term

Substitute the values into general formula \( a_1 = \sqrt{11} \), \( r = -\sqrt{11} \), and \( n = 9 \): \( a_9 = a_1 * r^{(9-1)} = \sqrt{11} * (-\sqrt{11})^{(8)} = 11^{4}.\)

Key Concepts

Common Ratio

Understanding the common ratio is crucial in the study of geometric sequences. In essence, the common ratio, typically denoted as \( r \), is the factor by which successive terms of a geometric sequence are multiplied. To find the common ratio, you simply divide one term by the previous term in the sequence. For instance, based on the given exercise, \( r = a_4/a_3 \). By calculating \( -11\sqrt{11}/11 \), we get the common ratio \( -\sqrt{11} \).

This ratio remains consistent throughout the sequence, and as such, each term can be generated by multiplying the previous term by the ratio. Knowing the common ratio is a key step in determining other properties of the sequence, such as future terms and the behavior of the sequence as it progresses.

Nth Term of a Sequence

When dealing with sequences, it's often necessary to find a specific term, known as the nth term. The nth term is essentially the value of an element that occupies the nth position in a sequence. For geometric sequences, the nth term can be calculated using the formula \( a_n = a_1 \times r^{(n-1)} \), where \( a_1 \) is the first term and \( r \) is the common ratio.

For example, to find the first term of the given geometric sequence, we rearrange the formula to compute \( a_1 = a_n / (r^{(n-1)}) \). With \( a_3 \) as 11, \( n \) as 3, and \( r \) as \( -\sqrt{11} \), we can find the first term, which is \( \sqrt{11} \). This process highlights the interconnected nature of the terms in a geometric sequence and the utility of the nth term formula.

Sequence and Series

A sequence is an ordered list of numbers following a particular pattern, while a series is the sum 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 the fixed non-zero number \( r \), the common ratio.

When we sum the terms of a geometric sequence, we create a geometric series. These concepts are foundational in mathematics, as they allow for the understanding of complex patterns and the ability to sum an infinite number of terms in certain conditions. For instance, in our exercise, we are asked to find the 9th term, which is part of the sequence. If we were to add up all terms from the first to the 9th, we would be discussing the series formed from this sequence.