Problem 8 - Precalculus: Mathematical 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: Understanding the Problem

We need to find the 4th term of a geometric sequence. We are given the first term \( a = \sqrt{3} \) and the common ratio \( r = \sqrt{3} \).

Step 2: General Formula for Geometric Sequence

The general formula for the \( n \)th term of a geometric sequence is \( a_n = a \cdot r^{n-1} \). To find the 4th term, we need to substitute \( n = 4 \) in the formula.

Step 3: Substitute Values into Formula

Substitute \( a = \sqrt{3} \), \( r = \sqrt{3} \), and \( n = 4 \) into the formula: \[a_4 = \sqrt{3} \cdot (\sqrt{3})^{4-1}.\]

Step 4: Simplify the Exponent

Simplify the exponent: \((\sqrt{3})^{3} = 3^{3/2} = 3 \times \sqrt{3} \) because \((\sqrt{3})^{2} = 3\) and multiplying by another \( \sqrt{3} \) gives the result.

Step 5: Calculate the 4th Term

Now calculate \( a_4 = \sqrt{3} \times 3 \times \sqrt{3} = 3 \times 3 = 9 \). Thus, the 4th term of the sequence is 9.

Key Concepts

nth term formula

In the realm of sequences, especially geometric sequences, the nth term formula is a cornerstone concept. It allows us to predict any term in a sequence without needing to know all the preceding terms. For a geometric sequence, the formula is given by:\[a_n = a \cdot r^{n-1}\]where:

• \(a_n\) denotes the nth term.
• \(a\) is the first term of the sequence.
• \(r\) is the common ratio, showing how each term is related to the previous one.
• \(n\) is the term number we are interested in.

Understanding this formula is crucial for working with geometric sequences effectively. It helps compute any term efficiently and is particularly useful when dealing with sequences of large lengths, where manual calculation might be impractical. Keep this formula handy, as it's your key to unlocking many sequence-related problems.

exponential growth

Exponential growth is a pattern of data that shows greater increases with passing time, creating a curve of growth when graphed. In the context of geometric sequences, each term grows by a consistent multiple, the common ratio. This type of sequence provides an excellent demonstration of exponential growth.Consider a situation where you start with a small number and keep multiplying by a fixed point, like the rule:

• If you start with 2 and double each time (common ratio of 2), you quickly jump from small numbers to massive ones.
• Starting at 3 and tripling takes it even faster!

The geometric sequence with a given first term and a constant common ratio naturally leads to exponential growth. This is displayed through the repeated multiplying factor in the nth term formula, \(a \cdot r^{n-1}\). The sequence grows faster as each term multiplies the previous one by \(r\). This rapid increase characterizes how exponential growth operates in mathematics and real-world scenarios.

sequence calculation

Calculating terms in a sequence, particularly a geometric one, involves substituting known values into the nth term formula and simplifying. With practice, this often straightforward process can become quick and intuitive.To calculate a specific term:

• Start with the given first term (\(a\)) and the common ratio (\(r\)).
• Decide the term position you are interested in (\(n\)).
• Substitute these values into the formula \(a_n = a \cdot r^{n-1}\).
• Simplify based on your values, such as handling exponents wisely.
• The result gives the value of the nth term directly.

Let's see an example with actual calculation: for the sequence given by the first term \(a = \sqrt{3}\), common ratio \(r = \sqrt{3}\), and interested in the 4th term:1. Substitute these values in the formula: \[a_4 = \sqrt{3} \cdot (\sqrt{3})^{4-1}\].2. Simplify the exponent: \((\sqrt{3})^3 = 3 \times \sqrt{3}\).3. Calculate: \(a_4 = 3 \times 3 = 9\).This process shows how structured sequence calculation becomes practical and powerful for determining any term in a sequence.