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Chapter 12: Problem 30

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

Short Answer

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

Step by step solution

01

Identify the First Term

The first term of the sequence is given as \( t \). So, \( a_1 = t \).
02

Find the Common Ratio

To find the common ratio \( r \), divide the second term by the first term: \[ r = \frac{\frac{t^2}{2}}{t} = \frac{t^2}{2} \cdot \frac{1}{t} = \frac{t}{2}. \]
03

Determine the Fifth Term

The general formula for the \( n\)th term of a geometric sequence is \( a_n = a_1 \cdot r^{n-1} \). For the fifth term, \( n=5 \): \[ a_5 = t \cdot \left( \frac{t}{2} \right)^4 = t \cdot \frac{t^4}{16} = \frac{t^5}{16}. \]
04

Find the General Formula for the \(n\)th Term

Substitute the first term and common ratio into the general formula for the \(n\)th term: \[ a_n = t \left( \frac{t}{2} \right)^{n-1}. \] Simplifying this, \[ a_n = t \cdot (t^{n-1}) \cdot \left(\frac{1}{2}\right)^{n-1} = \frac{t^n}{2^{n-1}}. \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Common Ratio
The common ratio is a fundamental part of understanding a geometric sequence. In a geometric sequence, each term is obtained by multiplying the previous term by a fixed, non-zero number known as the common ratio. Identifying the common ratio is crucial, as it is a key element in determining any term within the sequence.
\[ r = \frac{\text{second term}}{\text{first term}} \]
  • For the sequence provided: we have the first term as \( t \) and the second term as \( \frac{t^2}{2} \).
  • The common ratio, \( r \), is calculated by dividing the second term by the first term, yielding \( \frac{t}{2} \).
Understanding and calculating this ratio is essential in order to predict subsequent terms or to find a particular term in the sequence.
N-th Term Formula
The nth term formula is a powerful tool for determining any term in a geometric sequence. It enables you to find the term at any position \( n \) without having to list all previous terms.
The formula for the nth term of a geometric sequence is given by:
\[ a_n = a_1 \cdot r^{n-1} \]
  • Here, \( a_1 \) is the first term and \( r \) is the common ratio.
  • For our specific sequence, the first term \( a_1 \) is \( t \), and the common ratio \( r \) is \( \frac{t}{2} \).
  • By substituting into the formula, the nth term can be expressed as:
    \[ a_n = t \cdot \left( \frac{t}{2} \right)^{n-1} \]
  • After simplifying this, the formula becomes:
    \[ a_n = \frac{t^n}{2^{n-1}} \]
This formula allows you to compute any term \( a_n \) in the sequence directly.
Geometric Progression
A geometric progression (or geometric sequence) is a sequence in which each term is derived by multiplying the previous term by a fixed non-zero number, known as the common ratio. This type of sequence has unique properties distinct from an arithmetic progression, where terms differ by a fixed sum rather than a product.
In our example, the sequence starts with the term \( t \) and continues with noticeable patterns:
  • The second term is \( \frac{t^2}{2} \), and each successive term is obtained by multiplying by the common ratio \( \frac{t}{2} \).
  • This results in terms progressively having higher powers of \( t \) and increasing denominators by powers of 2.
Understanding a sequence as a geometric progression is key in predicting its behavior and solving related problems.
Sequence Terms
Sequence terms refer to the individual elements or values within a sequence. In a geometric sequence, these terms are crucial because each one is obtained through multiplication by the common ratio from its preceding term.
For each natural number \( n \), the terms of the sequence can be generated and are represented as \( a_n \).
  • The first few terms of our sequence are:
    • First term \( (a_1) = t \)
    • Second term \( (a_2) = \frac{t^2}{2} \)
    • Third term \( (a_3) = \frac{t^3}{4} \)
  • Each term is specifically calculated using the formula for a geometric sequence鈥檚 nth term.
Recognizing and constructing the sequence terms is foundational in comprehending the sequence鈥檚 overall structure and characteristics.

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