Problem 26 - College Algebra

Question

For the following exercises, write a recursive formula for each geometric sequence. $$ a_{n}=\\{14,56,224,896, \ldots\\} $$

Step-by-Step Solution

Answer

The recursive formula is \( a_{n} = 4 \times a_{n-1} \) with \( a_1 = 14 \).

Step 1: Identify the Pattern

First, we'll determine the pattern in the sequence to identify the common ratio, which is the factor by which each term is multiplied to get to the next term. Start by dividing the second term by the first term: \( \frac{56}{14} = 4 \). Now divide the third term by the second term: \( \frac{224}{56} = 4 \). Thus, the common ratio \( r \) is 4.

Step 2: Write the Recursive Formula

A recursive formula for a geometric sequence relates each term to the one before it. For a geometric sequence with a first term \( a_1 \) and a common ratio \( r \), the recursive formula is generally \( a_{n} = r imes a_{n-1} \) where \( n \geq 2 \). Knowing the first term, \( a_1 = 14 \), and the common ratio, \( r = 4 \), the recursive formula becomes \( a_{n} = 4 imes a_{n-1} \), with \( a_1 = 14 \).

Key Concepts

Geometric Sequence

A geometric sequence is a list of numbers where each term is generated by multiplying the previous term by a fixed number, known as the common ratio. This fixed number remains constant throughout the sequence, making it predictable and easy to follow. In essence, if you have the first term and the common ratio, you can determine all subsequent terms in the sequence. For example, given a sequence \( a_{n} = \{14, 56, 224, 896, \ldots\} \), we can see a clear pattern where each number is multiplied by 4 to get the next term.

• The sequence starts with 14,
• each following term is the previous term multiplied by 4.

Geometric sequences are very useful in mathematics, particularly when modeling exponential growth or decay. Understanding them can provide insight into numerous real-world phenomena.

Common Ratio

The common ratio in a geometric sequence is the constant factor that you multiply a term by to get the next term. It can be found by dividing any term in the sequence by its preceding term. Detecting this ratio is crucial because it dictates how the sequence progresses.Let's analyze the example \( a_{n} = \{14, 56, 224, 896, \ldots\} \). Here:

• Divide the second term (56) by the first (14) to get a common ratio of 4: \( \frac{56}{14} = 4 \).
• Verify this by dividing the third term (224) by the second (56): \( \frac{224}{56} = 4 \).
• Thus, the common ratio \( r \) is consistently 4.

This constant ratio is critical because it maintains the structure of the sequence, making calculations precise and replicable.

First Term

The first term in a geometric sequence is the starting point from which all subsequent terms are derived. It is often denoted as \( a_1 \) and can be considered the foundation of the sequence. In the realm of geometric sequences, the first term is crucial for setting the sequence in motion.For the sequence \( a_{n} = \{14, 56, 224, 896, \ldots\} \), the first term \( a_1 \) is 14.

• It acts as the base from which each subsequent term is calculated.
• By multiplying this first term by the common ratio, all future terms in the sequence can be generated.

Understanding the importance of the first term will aid in not only forming the sequence but also applying the recursive formula. Remember, knowing the first term and the common ratio allows for the complete recreation of the sequence.