Problem 51 - College Algebra

Question

Use recursive formulas to give two examples of geometric sequences whose \(3^{\text {rd }}\) terms are 200 .

Step-by-Step Solution

Answer

Sequences: (1) \( a_1 = 25, \ r = 2 \); (2) \( a_1 = 200, \ r = 1 \).

Step 1: Understanding the Recursive Formula for a Geometric Sequence

A geometric sequence is defined by the recursive formula \( a_n = a_{n-1} \times r \), where \( a_n \) is the \( n \)-th term, \( a_{n-1} \) is the previous term, and \( r \) is the common ratio.

Step 2: Establish the Base and Recursive Relationship

To find a sequence where the 3rd term is 200, we start with the tension that the general form for the first few terms is:\[ a_1 = a, \quad a_2 = ar, \quad a_3 = ar^2 \] where \( ar^2 = 200 \).

Step 3: Determine Possible Values for \( a_1 \) and \( r \)

Since \( ar^2 = 200 \), we need to express \( a_1 \) and \( r \) so that their product equals 200.Let's take \( a_1 = 25 \) and \( r = 2 \), then the sequence becomes: \( a_1 = 25, \ a_2 = 50, \ a_3 = 100 \times 2 = 200 \).

Step 4: Verify the First Example Sequence

We check if the third term for the sequence starting with \( a_1 = 25 \) and \( r = 2 \) indeed equals 200:\( a_1 = 25 \), \( a_2 = 25 \times 2 = 50 \), and \( a_3 = 50 \times 2 = 200 \). The sequence is correct.

Step 5: Choose Alternate Values for \( a_1 \) and \( r \)

Another example could start with \( a_1 = 200 \) and \( r = 1 \), where all terms are equal, so the third term is still 200. The sequence is \( a_1 = 200, \ a_2 = 200, \ a_3 = 200 \).

Step 6: Verify the Second Example Sequence

Verify if \( a_1 = 200 \) and \( r = 1 \) form a valid sequence:Each term equals the previous term multiplied by 1, making the sequence constant and the third term indeed 200.

Key Concepts

Recursive Formula

In the context of geometric sequences, understanding the recursive formula is crucial. It defines how each term in the series relates to its predecessor. Specifically, the formula for a geometric sequence is given by:

• \( a_n = a_{n-1} \times r \)

Here, \( a_n \) is the \( n \)-th term of the sequence, \( a_{n-1} \) is the term just before it, and \( r \) represents the common ratio.
This recursive formula allows us to easily generate the sequence term by term, starting from a given initial or base term. For example, if the first term \( a_1 \) is 25 and the common ratio \( r \) is 2, then to find the second term \( a_2 \), we multiply \( 25 \) by \( 2 \) to get \( 50 \). Similarly, to find the third term \( a_3 \), we multiply the second term \( 50 \) by \( 2 \) to get \( 100 \).
It is essential to establish a strong foundation with this formula as it is used to effortlessly propagate the sequence forward.

Common Ratio

The common ratio, denoted by \( r \), is the defining characteristic of a geometric sequence. This ratio is the factor by which we multiply each term to find the next term in the sequence.
In simpler terms, if you know one term and the common ratio, you can determine subsequent terms easily. For example, in the example of the sequence beginning with \( 25 \), using a common ratio of \( 2 \) produced a sequence of \( 25, 50, 100, 200, \) and so on.

• If the common ratio \( r \) is greater than \( 1 \), the sequence will grow exponentially.
• If \( r \) is between \( 0 \) and \( 1 \), the sequence will decay towards zero.
• If \( r = 1 \), the sequence remains constant.

Understanding the common ratio helps in predicting the behavior of the sequence whether it grows or shrinks as it progresses.

Geometric Progression

A geometric progression is simply a series of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number—the common ratio \( r \). This progression is not just confined to exercises in textbooks but is also a fundamental concept in various real-world scenarios such as finance (compounded interest), population growth, and even in physics.To identify a geometric sequence from other types of sequences, look for the pattern of multiplication. If you notice that each term can be obtained by multiplying the previous term by the same number, you're dealing with a geometric sequence.
In the exercise, a geometric sequence was needed where the third term equals 200. Using the understanding that the formula for a geometric sequence can take the form:

• \( a_3 = a_1 \times r^2 \)

We can calculate the necessary values for the first term \( a_1 \) and the common ratio \( r \) to achieve the specific requirement. For example, starting with \( a_1 = 25 \) and \( r = 2 \) creates a series that indeed makes \( a_3 = 200 \).
Grasping the principles of geometric progression aids in efficiently solving such sequence-related problems.