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Chapter 11: Problem 10

Find the indicated term of each geometric sequence. $$ a_{3}=24, r=\frac{1}{2}, n=7 $$

Short Answer

Expert verified
The 7th term of the sequence is \( \frac{3}{2} \).

Step by step solution

01

Identify the formula

The formula to find the nth term of a geometric sequence is given by \( a_n = a_1 \, r^{(n-1)} \), where \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the term number.
02

Express \( a_1 \) using \( a_3 \)

We are given \( a_3 = 24 \). Using the formula for the third term, we have \( a_3 = a_1 \, r^2 \). Substitute \( r = \frac{1}{2} \): \( 24 = a_1 \, \left( \frac{1}{2} \right)^2 \).
03

Solve for \( a_1 \)

The equation becomes \( 24 = a_1 \, \frac{1}{4} \). To solve for \( a_1 \), multiply both sides by 4: \( a_1 = 96 \).
04

Use \( a_1 \) to find \( a_7 \)

Now that we have \( a_1 = 96 \), use the formula for the nth term with \( n = 7 \): \( a_7 = 96 \, \left( \frac{1}{2} \right)^6 \).
05

Calculate \( (\frac{1}{2})^6 \)

Calculate \( \left( \frac{1}{2} \right)^6 = \frac{1}{64} \).
06

Complete the calculation of \( a_7 \)

Substitute back into the equation for \( a_7 \): \( a_7 = 96 \, \frac{1}{64} = \frac{96}{64} \). Simplify this to get \( a_7 = \frac{3}{2} \).

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Key Concepts

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

nth term formula
In a geometric sequence, each term is derived from the preceding one by multiplying by a constant called the common ratio. To find any term in the sequence, you use the nth term formula:
  • \( a_n = a_1 \cdot r^{(n-1)} \)
This formula is vital since it provides a way to compute any term's value in the sequence, as long as you know the first term \( a_1 \), the common ratio \( r \), and the term number \( n \). By substituting these into the formula, you can directly calculate the nth term without having to list out all preceding terms. This makes finding higher terms more efficient. Understanding this formula is key to solving problems involving geometric sequences.
common ratio
The common ratio \( r \) in a geometric sequence is the factor by which each term is multiplied to obtain the next term. It's consistent throughout the series and can be calculated as:
  • \( r = \frac{a_{n+1}}{a_n} \)
In the exercise we have, the common ratio is given as \( \frac{1}{2} \). This means each term is half of the previous term. The common ratio is crucial because it determines the rate at which the sequence increases or decreases. If \( r > 1 \), the sequence grows; if \( 0 < r < 1 \), it decreases. Recognizing and applying the common ratio helps seamlessly identify the pattern of the sequence.
first term
The first term \( a_1 \) in a geometric sequence is a foundation since it is the starting point from which all other terms are generated. Determining \( a_1 \) when it is not explicitly given requires using the information available about other terms. For instance, if you know a subsequent term and the common ratio,
  • Say \( a_3 = 24 \) with \( r = \frac{1}{2} \), then use \( a_3 = a_1 \cdot r^2 \).
Solving \( 24 = a_1 \cdot (\frac{1}{2})^2 \), rearrange to find \( a_1 = 96 \). This first term allows you to use the nth term formula effectively to find any other term in the sequence.
term calculation
Calculating a specific term in a geometric sequence involves substituting into the nth term formula. Once you have \( a_1 \), \( r \), and \( n \), plug these into the expression \( a_n = a_1 \cdot r^{(n-1)} \). Take the exercise example:
  • Calculate \( a_7 \) where \( a_1 = 96 \), \( r = \frac{1}{2} \), and \( n = 7 \).
Step through the formula to find \( a_7 = 96 \cdot (\frac{1}{2})^6 \). To simplify, manage the exponent first: \((\frac{1}{2})^6 = \frac{1}{64} \). Multiply to get \( a_7 = 96 \times \frac{1}{64} = \frac{3}{2} \). Precise term calculation is a repetitive yet straightforward process, enhancing your understanding with practice.

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Most popular questions from this chapter

CHALLENGE Explain why \(\frac{12 !}{7 ! 5 !}+\frac{12 !}{6 ! 6 !}=\frac{13 !}{7 ! 6 !}\) without finding the value of any of the expressions.

ARTHMETIC SERIES Use mathematical induction to prove the formula \(a_{1}+\left(a_{1}+d\right)+\left(a_{1}+2 d\right)+\cdots+\left[a_{1}+(n-1) d\right]=\frac{n}{2}\left[2 a_{1}+(n-1) d\right]\) for the sum of an arithmetic series.

Find the indicated term of each expansion. fourth term of \(\left(x+\frac{1}{3}\right)^{7}\)

Expand each power. $$ (2 a+b)^{6} $$

Expand each power. $$ (a-2)^{4} $$
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