Problem 26 - College Algebra with Corequisite Support

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 \(a_n = 4 \cdot a_{n-1}\), with \(a_1 = 14\).

Step 1: Identify the First Term

The first term of the sequence \(a_1\) is \(14\). We denote this as \(a_1 = 14\).

Step 2: Find the Common Ratio

To find the common ratio \(r\) of the sequence, divide the second term by the first term. Calculate \(r = \frac{56}{14} = 4\).

Step 3: Write the Recursive Formula

The recursive formula of a geometric sequence is defined by \(a_n = r \cdot a_{n-1}\). In this case, substitute \(r = 4\) and \(a_1 = 14\) to get the recursive formula: \(a_n = 4 \cdot a_{n-1}\) for \(n \geq 2\), with initial condition \(a_1 = 14\).

Key Concepts

Recursive Formula

A recursive formula is a method of defining a sequence where the next term is calculated from one or more of the previous terms. In the context of geometric sequences, this is particularly useful for generating terms without having to compute each one from scratch.
To construct a recursive formula for a geometric sequence, we need two key components:

• The common ratio, denoted as \(r\), which we'll dive into later, helps us determine how each term relates to the previous one.
• The initial term, noted as \(a_1\).

The general form of the recursive formula for a geometric sequence is:\[ a_n = r \cdot a_{n-1} \text{ for } n \geq 2 \]This formula means that any term \(a_n\) can be found by multiplying the previous term \(a_{n-1}\) by the common ratio \(r\). In our specific example, since the first term \(a_1\) is 14 and the common ratio \(r\) is 4, the formula becomes:\[ a_n = 4 \cdot a_{n-1} \text{ with } a_1 = 14 \]This setup allows one to easily compute the sequence by choosing a starting point and applying a consistent pattern.

Common Ratio

The common ratio is a fundamental element of any geometric sequence. It tells us by what factor each term is multiplied to get the next term. In order to determine this ratio, you take any term in the sequence and divide it by the previous term.
For example, in the sequence \( 14, 56, 224, 896, \ldots \), the common ratio is found by dividing the second term by the first term:\[ r = \frac{56}{14} = 4 \]This calculation reveals that each term is four times the previous term. The common ratio is essential because it defines the exponential growth or decay of the sequence.
Knowing the common ratio also makes it easy to identify or write the recursive formula, as it dictates how we progress from one term to the next. In our specific sequence, the consistent common ratio of 4 tells us that the growth is uniform and predictable.

Initial Term

The initial term of a geometric sequence, often denoted as \(a_1\), is the starting point of the sequence. It's crucial since it anchors all subsequent terms, serving as the base from which the sequence unfolds. In the exercise we're examining, the initial term is given as \(a_1 = 14\).
This initial term gets us off the ground with the recursive formula. It tells us where the doubling, tripling, or otherwise multiplication begins. In our example, from the term \(a_1 = 14\), we can apply the recursive formula by multiplying with the common ratio of 4 to get the next terms:

• \(a_2 = 4 \cdot 14 = 56\)
• \(a_3 = 4 \cdot 56 = 224\)
• \(a_4 = 4 \cdot 224 = 896\)

Remember, setting the initial term is akin to setting the scene for a play; it establishes the context from which the entire story, or in this case, sequence, flows.