Problem 31 - College Algebra with Corequisite Support

Question

For the following exercises, write the first eight terms of the sequence. \(a_{1}=\frac{1}{24}, \quad \mathrm{a}_{2}=1, \quad a_{n}=\left(2 a_{n-2}\right)\left(3 a_{n-1}\right)\)

Step-by-Step Solution

Answer

The first eight terms are: \(\frac{1}{24}, 1, \frac{3}{4}, \frac{9}{2}, \frac{81}{4}, \frac{2187}{4}, \frac{531441}{16}, \frac{6973568802}{64}.\)

Step 1: Identify the given terms

Start by acknowledging the initial terms provided in the sequence: \[ a_1 = \frac{1}{24} \text{ and } a_2 = 1. \] These are the first two terms of the sequence and will serve as our base for finding subsequent terms.

Step 2: Understand the recursive formula

The sequence is defined recursively for \( n \geq 3 \) as: \[ a_n = (2a_{n-2})(3a_{n-1}). \] This formula uses both of the two preceding terms \( a_{n-2} \) and \( a_{n-1} \) to calculate each new term \( a_n \).

Step 3: Calculate the third term \(a_3\)

Use the recursive formula to find \(a_3\): \[ a_3 = (2a_1)(3a_2) = \left(2 \times \frac{1}{24}\right)\left(3 \times 1\right) = \frac{1}{4} \times 3 = \frac{3}{4}. \]

Step 4: Calculate the fourth term \(a_4\)

Use the recursive formula to find \(a_4\): \[ a_4 = (2a_2)(3a_3) = (2 \times 1)(3 \times \frac{3}{4}) = 2 \times \frac{9}{4} = \frac{18}{4} = \frac{9}{2}. \]

Step 5: Calculate the fifth term \(a_5\)

Use the recursive formula to find \(a_5\): \[ a_5 = (2a_3)(3a_4) = \left(2 \times \frac{3}{4}\right)\left(3 \times \frac{9}{2}\right) = \frac{3}{2} \times \frac{27}{2} = \frac{81}{4}. \]

Step 6: Calculate the sixth term \(a_6\)

Use the recursive formula to find \(a_6\): \[ a_6 = (2a_4)(3a_5) = \left(2 \times \frac{9}{2}\right)\left(3 \times \frac{81}{4}\right) = 9 \times \frac{243}{4} = \frac{2187}{4}. \]

Step 7: Calculate the seventh term \(a_7\)

Use the recursive formula to find \(a_7\): \[ a_7 = (2a_5)(3a_6) = \left(2 \times \frac{81}{4}\right)\left(3 \times \frac{2187}{4}\right) = \frac{162}{4} \times \frac{6561}{4} = \frac{531441}{16}. \]

Step 8: Calculate the eighth term \(a_8\)

Use the recursive formula to find \(a_8\): \[ a_8 = (2a_6)(3a_7) = \left(2 \times \frac{2187}{4}\right)\left(3 \times \frac{531441}{16}\right) = \frac{4374}{4} \times \frac{1594323}{16} = \frac{6973568802}{64}. \]

Key Concepts

Sequence Initialization

When dealing with mathematical sequences, the first step often involves sequence initialization. This means setting up the starting terms of a sequence. In the provided exercise, the initialization involves specifying the first two terms as follows:

• \( a_1 = \frac{1}{24} \)
• \( a_2 = 1 \)

These initial terms are critical because they form the foundation upon which the rest of the sequence is built. Without these values, it would be impossible to calculate the subsequent terms. In essence, sequence initialization sets the stage for determining the entire sequence, making it a crucial part of understanding recursive sequences.

Recursive Formula

A recursive formula is a mathematical expression that defines each term of a sequence using the preceding terms. In the exercise, the recursive formula is given by:
\[a_n = (2a_{n-2})(3a_{n-1})\]This formula indicates that for any term \( a_n \) (where \( n \geq 3 \)), we need to multiply double the value of the term from two positions back \( a_{n-2} \) with triple the value of the immediate preceding term \( a_{n-1} \). A key aspect of recursive formulas is understanding how past terms influence future values, allowing us to iteratively calculate terms without explicitly knowing an explicit formula for \( a_n \). Understanding this concept is essential in tackling problems involving recursive sequences.

Term Calculation

Term calculation in a recursive sequence involves using the initial terms and the recursive formula to determine subsequent terms. For instance, in the example given, the third term \(a_3\) is calculated as:

• \( a_3 = (2a_1)(3a_2) = \left(2 \times \frac{1}{24}\right)\left(3 \times 1\right) = \frac{3}{4} \)

This process involves substituting known values (like \(a_1\) and \(a_2\)) into the recursive formula to find the next term. As you progress in the sequence, each new term relies on previously calculated terms, making the calculations cumulative. It’s important to carefully follow each computation step, ensuring correct arithmetic operations, as errors in early terms can propagate through the sequence.

Mathematical Sequences

Mathematical sequences are ordered lists of numbers defined by specific patterns or rules. They can be finite or infinite. The sequences may follow arithmetic, geometric, quadratic, or other complex patterns, depending on the sequence's purpose and construction. In this exercise, the focus is on recursive sequences, which define each term based on prior terms. An understanding of sequences is vital since such patterns are prevalent in various fields of mathematics and science. Being able to work with sequences allows us to model and solve real-world problems involving patterns and series, thereby enabling a systematic approach to complex problems.