Problem 31 - Algebra 2

Question

Write an equation for the nth term of each geometric sequence. $$ 64,16,4, \dots $$

Step-by-Step Solution

Answer

The nth term is \(a_n = 64 \cdot \left( \frac{1}{4} \right)^{n-1}\).

Step 1: Identify the First Term

The first term of the sequence is \(a_1 = 64\). This is the starting point for the sequence.

Step 2: Calculate the Common Ratio

To find the common ratio \(r\), divide the second term by the first term: \(r = \frac{16}{64} = \frac{1}{4}\). This represents the factor by which each term is multiplied to obtain the next term.

Step 3: Write the General Formula for the nth Term

The formula for the nth term of a geometric sequence is given by \(a_n = a_1 \cdot r^{n-1}\). Using our identified values, this becomes \(a_n = 64 \cdot \left( \frac{1}{4} \right)^{n-1}\).

Key Concepts

nth term

In a geometric sequence, the nth term is the term located at position n in the sequence. To find this term, we make use of a specific formula known as the geometric sequence formula. The nth term is important because it allows us to find any term in the sequence without calculating all preceding terms. If you know the first term and the common ratio, you can directly determine the nth term. This is particularly useful when dealing with large indices where manually computing each term would be impractical. The nth term for the given sequence 64, 16, 4,... can be calculated using the formula who’s variables are derived in the next sections.

common ratio

The common ratio in a geometric sequence is the constant factor that each term is multiplied by to get the next term. Finding the common ratio is crucial as it gives the sequence its distinct geometric characteristics.
**Calculating the Common Ratio**

• Choose any two consecutive terms from the sequence.
• Divide the second term by the first term.

In our sequence, the common ratio is calculated as \(r = \frac{16}{64} = \frac{1}{4}\). This indicates that each term is obtained by multiplying the previous term by \(\frac{1}{4}\). Understanding the common ratio helps us predict and compute any term in the sequence.

geometric sequence formula

The geometric sequence formula is the mathematical equation used to find any term in a geometric sequence. This formula is simple yet powerful. It requires two primary components: the first term of the sequence, denoted by \(a_1\), and the common ratio \(r\).
**General Formula**The formula for the nth term of a geometric sequence is:\[ a_n = a_1 \cdot r^{n-1} \]**Application**

• Identify the first term of the sequence, which in our case is 64.
• Calculate the common ratio, which we determined to be \(\frac{1}{4}\).
• Substitute these values into the formula to find the nth term.

For the sequence starting with 64, the nth term formula is:\[ a_n = 64 \cdot \left( \frac{1}{4} \right)^{n-1} \]This formula provides a way to find any term in the sequence efficiently, emphasizing the power and utility of understanding geometric sequences.