Problem 10 - Algebra 2

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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.

Step 2: 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 \( r = \frac{1}{2} \): \( 24 = a_1 \, \left( \frac{1}{2} \right)^2 \).

Step 3: Solve for \( a_1 \)

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

Step 4: Use \( a_1 \) to find \( a_7 \)

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

Step 5: Calculate \( (\frac{1}{2})^6 \)

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

Step 6: 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 \( a_7 = \frac{3}{2} \).

Key Concepts

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.