Problem 12 - Precalculus Mathematics for Calculus

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: Understand the Formula for the n-th Term

The n-th term of a geometric sequence can be found using the formula \( a_n = a \cdot r^{n-1} \). Here, \( a \) is the first term and \( r \) is the common ratio of the sequence.

Step 2: Plug in the Given Values into the Formula

We are given the first term \( a = \sqrt{3} \) and the common ratio \( r = \sqrt{3} \). We want to find the fourth term, so \( n = 4 \). Substitute these values into the formula: \[ a_4 = \sqrt{3} \cdot (\sqrt{3})^{4-1} \].

Step 3: Simplify the Expression

Simplify the expression: \[(\sqrt{3})^{4-1} = (\sqrt{3})^3 = \sqrt{3} \cdot \sqrt{3} \cdot \sqrt{3} = 3\sqrt{3} \]. Thus, \[ a_4 = \sqrt{3} \cdot 3\sqrt{3} \].

Step 4: Final Calculation

Calculate the final value: \[ a_4 = \sqrt{3} \cdot 3\sqrt{3} = 3(\sqrt{3} \cdot \sqrt{3}) = 3 \times 3 = 9 \].Therefore, the fourth term of the sequence is 9.

Key Concepts

n-th term formula

A geometric sequence is a series of numbers where each term increases or decreases by a constant factor known as the common ratio. To find any specific term in this sequence, we use the n-th term formula. The formula is written as \( a_n = a \cdot r^{n-1} \), where:

• \( a \) is the first term of the sequence
• \( r \) is the common ratio
• \( n \) is the term number you want to find

For example, if we have a first term \( a = \sqrt{3} \) and a common ratio \( r = \sqrt{3} \), you can find any term in the sequence by plugging these values into the formula. For the fourth term as an example, you would calculate \( a_4 = \sqrt{3} \cdot (\sqrt{3})^{4-1} \). This formula gives you a way to quickly find any term in the sequence without listing all the previous terms.

common ratio

The common ratio is a crucial element of a geometric sequence and is denoted by \( r \). It defines how each term in the sequence is related to the previous term. To find the common ratio, divide any term in the sequence by its preceding term. If every new term is produced by multiplying the previous one by the same value, that value is your common ratio.
For instance, with \( r = \sqrt{3} \), each term in the sequence is multiplied by \( \sqrt{3} \) to get the next term. This characteristic makes geometric sequences unique, as the change between each term is exponential, not linear. Understanding the common ratio helps you gauge how quickly a sequence grows or shrinks.

fourth term calculation

Calculating the fourth term of a geometric sequence involves using both the n-th term formula and the common ratio. For the given sequence, you plug the values \( a = \sqrt{3} \) and \( r = \sqrt{3} \) into the formula: \( a_4 = \sqrt{3} \cdot (\sqrt{3})^{4-1} \).
When you simplify \( (\sqrt{3})^{3} \), you multiply \( \sqrt{3} \cdot \sqrt{3} \cdot \sqrt{3} = 3\sqrt{3} \). Thus, the expression becomes \( a_4 = \sqrt{3} \cdot 3\sqrt{3} \). Further simplification gives \( 3(\sqrt{3} \cdot \sqrt{3}) = 3 \times 3 = 9 \).
So, by following these calculations, you find that the fourth term of the sequence is 9. This step-by-step approach ensures clarity in how the formula is applied and showcases the power of the common ratio in altering each term of the sequence exponentially.