Problem 30 - Precalculus: Mathematical for Calculus

Question

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

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 common ratio

A geometric sequence has a common ratio between consecutive terms. Given the first term as \( t \) and the second term as \( \frac{t^2}{2} \), find the common ratio by dividing the second term by the first term: \[ r = \frac{\frac{t^2}{2}}{t} = \frac{t^2}{2} \times \frac{1}{t} = \frac{t}{2}.\]

Step 2 - Find the fifth term

Each term in a geometric sequence is found by multiplying the previous term by the common ratio. To find the fifth term, multiply the fourth term \( \frac{t^4}{8} \) by the common ratio \( \frac{t}{2} \):\[t_5 = \frac{t^4}{8} \times \frac{t}{2} = \frac{t^5}{16}.\]

Step 3 - Derive the formula for the nth term

The nth term formula for a geometric sequence is given by \(a_n = a_1 \times r^{n-1}\). Here, \(a_1 = t\) and \(r = \frac{t}{2} \). Substitute these values into the formula:\[a_n = t \times \left(\frac{t}{2}\right)^{n-1} = \frac{t^n}{2^{n-1}}.\]

Key Concepts

Common Ratio

In a geometric sequence, understanding the concept of a "common ratio" is crucial. The common ratio is the factor between any two consecutive terms in the sequence. You find it by dividing a term by the previous term.
For example, in the sequence given:

• First term: \( t \)
• Second term: \( \frac{t^2}{2} \)

To find the common ratio \( r \), divide the second term by the first term:\[ r = \frac{\frac{t^2}{2}}{t} = \frac{t}{2} \].
This means each term is \( \frac{t}{2} \) times the previous term. Knowing this ratio helps you move from one term to the next in the sequence.

Nth Term Formula

The nth term formula is a tool that allows you to find any term in a geometric sequence without listing all the terms. It's like a shortcut! The formula for the nth term \( a_n \) is:\[ a_n = a_1 \times r^{n-1} \], where:

• \( a_1 \) is the first term
• \( r \) is the common ratio
• \( n \) is the term number

In our sequence, the first term \( a_1 \) is \( t \), and the common ratio \( r \) is \( \frac{t}{2} \). Plug these into the formula to get:\[ a_n = t \times \left(\frac{t}{2}\right)^{n-1} = \frac{t^n}{2^{n-1}} \].
This formula is very useful, especially for finding terms far into the sequence without having to calculate each term individually up to that point.

Geometric Progression

A geometric progression is essentially a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. This structure is predictable and grows exponentially.
Let's look at a quick example: if the first term is 6 and the common ratio is 3, the sequence would be:

• First term: 6
• Second term: \( 6 \times 3 = 18 \)
• Third term: \( 18 \times 3 = 54 \)

This regularity is what makes geometric progressions so interesting as they lend themselves well to algebraic manipulation and provide clear, pattern-based examples. Knowing how to handle a geometric progression can help solve many problems in mathematics, finance, and even computer science.

Sequence Term Calculation

Calculating terms in a geometric sequence can seem daunting at first, but it becomes simple once you understand the common ratio and nth term formula.
To calculate a specific term \( a_n \), follow these steps:

• Identify the first term \( a_1 \).
• Calculate the common ratio \( r \).
• Use the nth term formula: \( a_n = a_1 \times r^{n-1} \).
• Substitute your known values into the formula.

For instance, if you want to find the fifth term as in the original exercise:

• The fourth term is \( \frac{t^4}{8} \).
• Multiply by the common ratio \( \frac{t}{2} \) to get: \[ t_5 = \frac{t^4}{8} \times \frac{t}{2} = \frac{t^5}{16} \].

This systematic approach ensures accuracy and efficiency in finding any term in the sequence.