Problem 21 - College Algebra

Question

For the following exercises, write an explicit formula for each sequence. $$ 4,7,12,19,28, \ldots $$

Step-by-Step Solution

Answer

The explicit formula is \( a_n = n^2 + 3 \).

Step 1: Identify the Pattern

Examine the differences between consecutive terms in the sequence: \( 7 - 4 = 3 \), \( 12 - 7 = 5 \), \( 19 - 12 = 7 \), and \( 28 - 19 = 9 \). Notice that the differences increase by 2 each time.

Step 2: Recognize the Pattern Type

The pattern of increasing differences indicates a quadratic sequence, which can be expressed in the form \( a_n = an^2 + bn + c \).

Step 3: Set Up Equations for Initial Terms

Use the first few terms to set up equations: \( a_1 = 4 \), \( a_2 = 7 \), \( a_3 = 12 \). Substitute these into the quadratic formula to form equations:1. \( a(1)^2 + b(1) + c = 4 \)2. \( a(2)^2 + b(2) + c = 7 \)3. \( a(3)^2 + b(3) + c = 12 \).

Step 4: Solve the System of Equations

Simplify each equation:1. \( a + b + c = 4 \)2. \( 4a + 2b + c = 7 \)3. \( 9a + 3b + c = 12 \).Subtract equation (1) from (2) and (3) to find \( a \) and \( b \). Then solve for \( c \):- Subtract (1) from (2): \( 3a + b = 3 \)- Subtract (2) from (3): \( 5a + b = 5 \).Subtract these results to find \( a \):- \( 5a + b - (3a + b) = 5 - 3 \)- \( 2a = 2 \) which implies \( a = 1 \).Substitute \( a = 1 \) into \( 3a + b = 3 \):- \( 3(1) + b = 3\)- \( b = 0 \).Now solve for \( c \) using equation (1) again with known values for \( a \) and \( b \):- \( 1 + 0 + c = 4 \)- \( c = 3 \).

Step 5: Write the Explicit Formula

With the values found, the explicit formula for the sequence is \( a_n = n^2 + 3 \).

Key Concepts

Explicit Formula

When exploring any sequence, one of the key objectives is to find a concise mathematical expression that captures the pattern of that sequence. This expression is known as the explicit formula. An explicit formula provides a direct method to calculate the nth term of the sequence, making it significantly easier than computing each term through recursive steps.

For quadratic sequences, which involve terms that fit a specific parabolic distribution, the pattern can be captured using a formula of the form:

• \( a_n = an^2 + bn + c \)

Here, \( a \), \( b \), and \( c \) are constants that we can find through a set of equations derived from the given terms of the sequence. This explicit formula is very powerful, as it directly links each term's position in the sequence with its value.

Once you have determined the constants \( a \), \( b \), and \( c \), you can express any term in the sequence with a simple calculation. This is the beauty and utility of the explicit formula—it turns a complex and potentially endless sequence into something manageable and predictable.

Pattern Recognition

Recognizing the underlying structure or pattern is essential when working with sequences. In the case of quadratic sequences, the differences between consecutive terms don't remain constant but instead increase in a regular pattern.

Initially, you might calculate the first-level differences, which don't yield a steady sequence, indicating the sequence is not linear. Therefore, you’ll need to look at the second-level differences, revealing a uniform pattern.

For example:

• The differences are calculated as \(3, 5, 7, 9 \)... with each value showing a consistent increase of +2.

Discovering this change in differences is the key to identifying that you are dealing with a quadratic sequence. Once identified, pattern recognition allows you to proceed with setting up the proper explicit formula model for further analysis.

This skill of pattern recognition is invaluable across many aspects of mathematics, enabling you to simplify and solve many types of problems, especially those involving complex sequences.

System of Equations

To find the explicit formula's constants in a quadratic sequence, after identifying the pattern, you need to solve a system of equations. This method involves using a few initial terms of the sequence and substituting them into the quadratic formula.

You start by equating these terms to the general formula \( a_n = an^2 + bn + c \), forming a series of equations:

• \( a(1)^2 + b(1) + c = 4 \)
• \( a(2)^2 + b(2) + c = 7 \)
• \( a(3)^2 + b(3) + c = 12 \)

Simplifying these gives you a system of equations that can be solved to find the values of \( a \), \( b \), and \( c \). By solving this system step by step, you can determine the constants:

• Subtract equations to eliminate variables and simplify.
• Find each constant, enabling you to compile the final explicit formula.

Understanding how to construct and solve these systems is crucial not only for sequences but also for broader applications in algebra and calculus. It teaches a structured approach to solving complex mathematical problems by breaking them down into simpler, more manageable parts.