Problem 65 - College Algebra with Corequisite Support

Question

List the first four terms of the sequence \(a_{n}=5.7^{n}+0.275(n-1) !\)

Step-by-Step Solution

Answer

The first four terms are 35.275, 245.275, 1715.55, and 12006.65.

Step 1: Understand the Sequence Formula

The sequence is given by the formula \(a_{n} = 5\cdot7^{n} + 0.275\cdot(n-1)! \). This formula includes both an exponential term \(7^{n}\) and a factorial term \((n-1)!\). To find the sequence terms, these components must be calculated separately for each \(n\).

Step 2: Calculate the First Term \(a_{1}\)

For \(n=1\): - The factorial part is \((1-1)! = 0! = 1\).- Substitute into the formula: \(a_{1} = 5\cdot7^{1} + 0.275\cdot1 = 5\cdot7 + 0.275 = 35 + 0.275 = 35.275\).

Step 3: Calculate the Second Term \(a_{2}\)

For \(n=2\):- The factorial part is \((2-1)! = 1! = 1\).- Substitute into the formula: \(a_{2} = 5\cdot7^{2} + 0.275\cdot1 = 5\cdot49 + 0.275 = 245 + 0.275 = 245.275\).

Step 4: Calculate the Third Term \(a_{3}\)

For \(n=3\):- The factorial part is \((3-1)! = 2! = 2\).- Substitute into the formula: \(a_{3} = 5\cdot7^{3} + 0.275\cdot2 = 5\cdot343 + 0.55 = 1715 + 0.55 = 1715.55\).

Step 5: Calculate the Fourth Term \(a_{4}\)

For \(n=4\):- The factorial part is \((4-1)! = 3! = 6\).- Substitute into the formula: \(a_{4} = 5\cdot7^{4} + 0.275\cdot6 = 5\cdot2401 + 1.65 = 12005 + 1.65 = 12006.65\).

Key Concepts

Understanding Factorials

The concept of factorials is fundamental in sequences involving permutations and combinations. The factorial of a number, denoted as \(n!\), is the product of all positive integers less than or equal to \(n\). This means that \(n! = n \times (n-1) \times (n-2) \times ... \times 1\). Factorials are frequently used in sequences to compute the terms that involve decreasing multiplication, effectively scaling the values based on the given pattern. In our original exercise, the formula for the sequence involves \((n-1)!\), so we calculate the factorial of one less than each term, \(n\). This affects how quickly the terms' values grow as \(n\) increases. One particular instance to remember is that \(0!\) is defined as \(1\), a special case often used in mathematical reasoning.
Here’s a quick list of small number factorials:

• \(0! = 1\)
• \(1! = 1\)
• \(2! = 2 \times 1 = 2\)
• \(3! = 3 \times 2 \times 1 = 6\)
• \(4! = 4 \times 3 \times 2 \times 1 = 24\)

Mastering factorials is a key step in solving sequence-related problems effectively.

Exploring the Exponential Function

An exponential function is a mathematical expression in which a quantity is raised to a power, typically involving the notation \(a^x\) where \(a\) is a constant base and \(x\) is the exponent. In the sequence formula \(a_{n} = 5 \cdot 7^n + 0.275 \cdot (n-1)!\), the exponential component \(7^n\) is a significant contributor. Here, \(7\) is the base, and \(n\) changes as each sequence term is calculated.
The exponential function contributes massively to the growth of terms as \(n\) increases, making the values increase at a rapid rate. This is evident in our sequence, where you can observe how swiftly \(7^n\) escalates through the terms. For example:

• At \(n = 1\), \(7^1 = 7\).
• At \(n = 2\), \(7^2 = 49\).
• At \(n = 3\), \(7^3 = 343\).

Recognizing the exponential function's role in sequences helps in predicting how quickly terms are going to increase and estimating the magnitude of sequence terms for higher indices.

Sequence Terms Breakdown

Sequence terms are individual components derived from a sequence formula for distinct values of \(n\). Each term is calculated independently based on the sequence's structure. The original sequence formula we are examining has highlighted both exponential and factorial components, which are combined to produce each term. The process generally involves calculating each part separately and then combining them according to the sequence's rules.
In our exercise, for finding the first four terms, we computed:

• \(a_1 = 5 \cdot 7^1 + 0.275 \cdot 0! = 35 + 0.275 = 35.275\)
• \(a_2 = 5 \cdot 7^2 + 0.275 \cdot 1! = 245 + 0.275 = 245.275\)
• \(a_3 = 5 \cdot 7^3 + 0.275 \cdot 2! = 1715 + 0.55 = 1715.55\)
• \(a_4 = 5 \cdot 7^4 + 0.275 \cdot 3! = 12005 + 1.65 = 12006.65\)

Each term is the result of an interaction between the exponential function and the factorial, demonstrating how numerical sequences can model complex phenomena with relative simplicity.