Problem 8 - Algebra and Trigonometry

Question

Find the \(n\)th term of the geometric sequence with given first term \(a\) and common ratio \(r .\) What is the fourth term? $$ a=\sqrt{3}, \quad r=\sqrt{3} $$

Step-by-Step Solution

Answer

The fourth term is 9.

Step 1: Understanding the Geometric Sequence Formula

The formula for the nth term of a geometric sequence is given by \( a_n = a imes r^{(n-1)} \), where \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the term number we are looking for.

Step 2: Substituting Given Values

We substitute the given values into the formula: \( a = \sqrt{3} \), \( r = \sqrt{3} \), and we need the 4th term (\( n = 4 \)). This gives us \( a_4 = \sqrt{3} \times (\sqrt{3})^{4-1} \).

Step 3: Calculating the Exponent

Simplify the exponent part first: \( (\sqrt{3})^{3} = (3^{0.5})^3 = 3^{1.5} = 3 \times \sqrt{3} \).

Step 4: Multiplying to Find the Fourth Term

Substitute back into the formula: \( a_4 = \sqrt{3} \times 3 \times \sqrt{3} \). Simplify the multiplication step: \( \sqrt{3} \times \sqrt{3} = 3 \), so \( a_4 = 3 \times 3 = 9 \).

Key Concepts

The nth Term Formula in Geometric Sequences

The formula to find any term in a geometric sequence is crucial for understanding how sequences progress. The formula is: \[a_n = a \times r^{(n-1)}\] Here, \(a\) represents the first term, \(r\) is the common ratio, and \(n\) is the term number you are calculating.
This powerful formula allows you to find the value of any term based on its position in the sequence.Let’s break it down:

• \(a\): First term – it’s the starting point of your sequence.
• \(r^{(n-1)}\): Shows how many times you multiply the common ratio to the first term.

Understanding this formula helps you identify patterns and predict future values in the sequence.
It is particularly useful in various real-world examples, such as calculating interest and modeling exponential growth.

Understanding the Common Ratio

A common ratio in a geometric sequence is the constant factor between any two consecutive terms. This ratio is consistent throughout the sequence and helps us understand the relationship between terms. If you multiply any term by the common ratio, you get the next term. For example, if the common ratio \(r = \sqrt{3}\):

• Start with the first term \(a = \sqrt{3}\).
• Multiply by \(r\) to get the next term.
• This pattern continues for all subsequent terms.

In our specific instance, the common ratio is \(\sqrt{3}\).
This means each term is the product of the previous term and \(\sqrt{3}\).
Recognizing a common ratio's role is essential to understanding the exponential nature of geometric sequences.

Identifying the First Term of a Sequence

The first term of a sequence, denoted as \(a\), acts as the foundation for the entire sequence. It sets the stage for what follows and is vital for any computation involving the sequence. In the given example, \(a = \sqrt{3}\). Knowing the initial term helps to determine the specific sequence:

• It is the starting point from which you develop additional terms using the common ratio.
• All formula calculations pivot around this initial term.
• It provides a concrete point of reference, needed for accurate term mapping.

Without identifying and understanding this first term, the sequence would lack direction and clarity. Thus, always begin your sequence calculations by confirming \(a\).