Problem 34 - Precalculus Mathematics for Calculus

Question

Determine the common ratio, the fifth term, and the \(n\) th term of the geometric sequence. $$t, \frac{t^{2}}{2}, \frac{t^{3}}{4}, \frac{t^{4}}{8}, \dots$$

Step-by-Step Solution

Answer

Common ratio is \(\frac{t}{2}\), fifth term is \(\frac{t^5}{16}\), \(n\)th term is \(\frac{t^n}{2^{n-1}}\).

Step 1: Identify the First Term

The first term of the geometric sequence is given as the initial term, which is \( t \). This is denoted as \( a_1 = t \).

Step 2: Calculate the Common Ratio

In a geometric sequence, the common ratio \( r \) is found by dividing any term by its preceding term. Calculate \( r \) using the first two terms: \[ r = \frac{\frac{t^2}{2}}{t} = \frac{t}{2} \].

Step 3: Find the Fifth Term

The general formula for the \( n \)th term in a geometric sequence is \( a_n = a_1 \cdot r^{n-1} \). To find the fifth term, substitute \( a_1 = t \), \( r = \frac{t}{2} \), and \( n = 5 \): \[ a_5 = t \cdot \left(\frac{t}{2}\right)^{4} = t \cdot \frac{t^4}{16} = \frac{t^5}{16} \].

Step 4: Derive the General Formula for the n-th Term

Using the general formula, substitute \( a_1 = t \) and the common ratio \( r = \frac{t}{2} \). This gives us the \( n \)th term: \[ a_n = t \left( \frac{t}{2} \right)^{n-1} = \frac{t^n}{2^{n-1}} \].

Key Concepts

Understanding the Common Ratio

In a geometric sequence, the common ratio is a crucial component that defines how the sequence progresses from one term to the next. It is the factor by which each term is multiplied to get the subsequent term. To find this common ratio, simply take any term in the sequence, and divide it by its preceding term. This will remain consistent throughout the sequence. For the given sequence \( t, \frac{t^{2}}{2}, \frac{t^{3}}{4}, \frac{t^{4}}{8}, \ldots \), the common ratio is found by dividing the second term \( \frac{t^{2}}{2} \) by the first term \( t \). Thus,

• \( r = \frac{\frac{t^2}{2}}{t} = \frac{t}{2} \).

The common ratio being \( \frac{t}{2} \) indicates that each subsequent term is the result of multiplying the previous term by \( \frac{t}{2} \). Understanding this ratio is the key to navigating through the terms of the sequence effectively.

Using the n-th Term Formula

The n-th term formula in a geometric sequence provides a direct way to compute any term in the sequence without calculating all the preceding terms. The formula is given by \( a_n = a_1 \cdot r^{n-1} \), where \( a_1 \) is the first term and \( r \) is the common ratio. For our sequence, with \( a_1 = t \) and \( r = \frac{t}{2} \), the n-th term, \( a_n \), can be calculated as:

• \( a_n = t \cdot \left( \frac{t}{2} \right)^{n-1} = \frac{t^n}{2^{n-1}} \).

This simple formula allows us to find the position of any term in the sequence quickly. You don't have to manually multiply each term up to \( n \). Instead, plug \( n \) directly into the formula for an efficient computation.

Calculating the Fifth Term of the Sequence

Finding a specific term in a sequence, such as the fifth term, can be made straightforward using the n-th term formula. With the formula \( a_5 = a_1 \cdot r^{4} \) and knowing that \( a_1 = t \) and \( r = \frac{t}{2} \), we can compute the fifth term as follows:

• \( a_5 = t \cdot \left( \frac{t}{2} \right)^4 = t \cdot \frac{t^4}{16} = \frac{t^5}{16} \).

This computation tells us that the fifth term of the sequence is \( \frac{t^5}{16} \). This method showcases how the n-th term formula simplifies the process, saving you time and ensuring precision.