Question

Write a rule for the geometric sequence with the given description. a. The first term is \(-3\), and each term is 5 times the previous term. b. The first term is 72 , and each term is \(\frac{1}{3}\) times the previous term.

Short answer

a. The rule for the sequence is \(a_n = -3 \cdot 5^{(n-1)}\). b. The rule for the sequence is \(a_n = 72 \cdot (1/3)^{(n-1)}\).

Step-by-step solution

Identify the first term and the common ratio in the sequence (Part a)

In Part a, the first term, \(a_1\) is -3 and the common ratio, \(r\), is 5.

Write the rule for the sequence (Part a)

The rule for a geometric sequence is \(a_n = a_1 \cdot r^{(n-1)}\). Substitute -3 for \(a_1\) and 5 for \(r\). Thus, the rule for this sequence is \(a_n = -3 \cdot 5^{(n-1)}\).

Identify the first term and the common ratio in the sequence (Part b)

In Part b, the first term, \(a_1\) is 72 and the common ratio, \(r\), is 1/3.

Write the rule for the sequence (Part b)

The rule for a geometric sequence is \(a_n = a_1 \cdot r^{(n-1)}\). Substitute 72 for \(a_1\) and 1/3 for \(r\). Thus, the rule for this sequence is \(a_n = 72 \cdot (1/3)^{(n-1)}\).

Key concepts

Common Ratio

In a geometric sequence, the common ratio is a key component. It's the constant factor by which each term in the sequence is multiplied to get the next term. This makes the sequence grow (or shrink) in a predictable pattern.

• To find the common ratio, you divide any term in the sequence by the previous term, provided the sequence is geometric.
• In the given problem, for part (a), the common ratio is 5, which means each term is 5 times the preceding term.
• For part (b), the common ratio is \( \frac{1}{3} \). Here, every term is a third of the previous one.

Understanding the common ratio helps you determine how the sequence behaves over time. If the ratio is greater than 1, the sequence increases, while a ratio between 0 and 1 indicates the sequence is decreasing. A negative ratio flips the sequence's direction with each term.

First Term

The first term of a geometric sequence, often represented as \( a_1 \), serves as the starting point of the sequence. It is crucial because the whole sequence is a result of stretching or shrinking this initial term through multiplication by the common ratio.

• In part (a) of the exercise, the first term \( a_1 \) is \(-3\). This means the sequence begins at \(-3\) and each subsequent term gets multiplied by the common ratio of 5.
• In part (b), \( a_1 \) is 72. Each following term is calculated by multiplying the previous term by \( \frac{1}{3} \).

Remember, the first term sets the tone of your sequence. By knowing this term and the common ratio, you can predict any term in the sequence.

Sequence Rule

The sequence rule is the formulation that allows for the calculation of any term in a geometric sequence without having to list all preceding terms. It typically uses the formula:\[ a_n = a_1 \cdot r^{(n-1)} \]where:

• \(a_n\) is the nth term we're solving for,
• \(a_1\) is the first term of the sequence,
• \(r\) is the common ratio,
• \(n\) is the position of the term within the sequence.

For part (a) of our exercise, substituting \( a_1 = -3 \) and \( r = 5 \) into the formula, we get: \[ a_n = -3 \cdot 5^{(n-1)} \]For part (b), using \( a_1 = 72 \) and \( r = \frac{1}{3} \), the rule becomes: \[ a_n = 72 \cdot \left( \frac{1}{3} \right)^{(n-1)} \]Knowing the sequence rule lets you compute any term directly. It's a helpful tool that reveals the sequence's structure and guides predictions of future terms.