Problem 34 - Algebra and Trigonometry

Question

Determine the common ratio, the fifth term, and the nth 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: \( \frac{t}{2} \), Fifth term: \( \frac{t^5}{16} \), nth term: \( \frac{t^n}{2^{n-1}} \).

Step 1: Identify the first term

The first term of the sequence is directly given as the first element in the sequence. Therefore, the first term is \( a_1 = t \).

Step 2: Determine the common ratio

To find the common ratio \( r \) in a geometric sequence, divide the second term by the first term. The second term is \( \frac{t^2}{2} \), so the common ratio is \( r = \frac{\frac{t^2}{2}}{t} = \frac{t}{2} \).

Step 3: Find the fifth term

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

Step 4: Determine the general formula for the nth term

The general formula for the \( n \)-th term is given by \( a_n = a_1 \cdot r^{n-1} \). Substituting \( a_1 = t \) and \( r = \frac{t}{2} \), we get: \[ a_n = t \cdot \left(\frac{t}{2}\right)^{n-1} = t \cdot \frac{t^{n-1}}{2^{n-1}} = \frac{t^n}{2^{n-1}} \]. So the \( n \)-th term is \( \frac{t^n}{2^{n-1}} \).

Key Concepts

Understanding the Common Ratio

In a geometric sequence, the common ratio is a key feature that defines how the sequence progresses from one term to the next. It is essentially the factor by which we multiply one term to receive the following term. To determine the common ratio, you choose any two consecutive terms and divide the second term by the first. This common ratio remains constant throughout the sequence.

Let's take a glance at our sequence: the first term is \( t \), and the second term is \( \frac{t^2}{2} \). By dividing the second term by the first, we find the common ratio:

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

This common ratio \( \frac{t}{2} \) tells us that each term is multiplied by \( \frac{t}{2} \) to produce the next term in the sequence. This factor plays a crucial role in all other calculations within the sequence.

Calculating the nth Term Formula

The nth term formula of a geometric sequence provides a way to find any term within the sequence without having to compute all the preceding terms individually. It is especially useful for a very large number of terms. The formula expresses the nth term as a product of the first term and the common ratio raised to the power of \( n-1 \).

This sequence has a first term of \( t \) and a common ratio of \( \frac{t}{2} \). By plugging these values into the nth term formula \( a_n = a_1 \cdot r^{n-1} \), we get:

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

This formula allows us to calculate any term of the sequence with ease. For example, if you want to find the 10th term, substitute 10 for \( n \) in the formula.

Finding the Fifth Term

To find the fifth term of a geometric sequence, you'll use the nth term formula and substitute \( n = 5 \). From the previous section, we know that our sequence follows the formula \( a_n = \frac{t^n}{2^{n-1}} \). By substituting \( n = 5 \), it becomes:

• \( a_5 = \frac{t^5}{2^{4}} = \frac{t^5}{16} \)

Breaking it down step-by-step, this means you took the first term \( t \) and multiplied it by the common ratio \( \frac{t}{2} \) raised to the fourth power. This approach ensures a quick and clear calculation, reaffirming that the fifth term is \( \frac{t^5}{16} \). So, if you're working with a test or exercise, you can quickly find this value without calculating every intermediate term.