Problem 24 - Algebra 2

Question

Decide whether each formula is explicit or recursive. Then find the first five terms of each sequence. $$ a_{n}=2 a_{n-1}+3, \text { where } a_{1}=3 $$

Step-by-Step Solution

Answer

The formula is recursive, and the first five terms are 3, 9, 21, 45, and 93

Step 1: Identify whether the formula is explicit or recursive

The formula \(a_{n}=2 a_{n-1}+3\) is a recursive formula as it defines each term in terms of its predecessor.

Step 2: Determine the first term

The first term \(a_{1}\) of the sequence is given as 3.

Step 3: Find the second term

Plug in \(n=2\) into the formula to find the second term. \(a_{2}=2 a_{1}+3 = 2(3) + 3 = 9\)

Step 4: Find the third term

Use the formula and plug in \(n=3\) to find the third term. \(a_{3}=2 a_{2}+3 = 2(9) + 3 = 21\)

Step 5: Find the fourth term

Using the formula to find the fourth term when \(n=4\).\(a_{4}=2 a_{3}+3 = 2(21) + 3 = 45\)

Step 6: Find the fifth term

Finally, find the fifth term \(a_{5}\) using the formula. \(a_{5}=2 a_{4}+3 = 2(45) + 3 = 93\)

Key Concepts

Sequence Terms

Sequence terms are the distinct elements or numbers that make up a sequence. In our case, each number in the sequence is derived from a specific rule or formula. Since sequences are ordered collections, the order of the terms is crucial. Each term is usually denoted by a subscript that indicates its position. For example, in the sequence given in the exercise:

• The first term is denoted as \(a_1\)
• The second term is represented by \(a_2\)
• This continues onward through \(a_3, a_4\), and so forth.

The terms serve as the fundamental building blocks of any sequence, forming the sequence when arranged in their specific order according to the given rule.

Recursive Formula

A recursive formula is a mathematical expression that determines each term in a sequence based on one or more preceding terms. This type of formula is advantageous when each term relates tightly to the one before it, allowing for a dynamic build-up of the sequence. In the provided example, the recursive formula is:\[ a_{n} = 2a_{n-1} + 3 \]This means that to calculate any term \(a_n\), you must incorporate the previous term \(a_{n-1}\) into the formula. The sequence begins with the initial term \(a_1\), given as 3. From there:

• To find \(a_2\), use \(a_1\)
• To find \(a_3\), use \(a_2\)
• This pattern continues for subsequent terms.

A recursive formula requires knowing at least one initial term to generate the sequence, as each calculation builds off the previous result.

Explicit Formula

Unlike the recursive formula, an explicit formula expresses each term of a sequence independently of the others. This type of formula allows you to plug in the position number to get the term directly, without referencing previous terms. Unfortunately, our example sequence does not initially provide an explicit formula. For sequences with explicit formulas, they often appear like:\[ a_n = f(n) \] This formula could directly compute the term without prior calculations. This can be incredibly efficient for finding specific terms far along in the sequence without computing all preceding terms. Essentially, an explicit formula offers a straightforward way to jump directly to any term in the sequence.

Mathematical Sequences

Mathematical sequences are ordered lists of numbers defined by specific rules. These rules can be represented as either:

• Recursive formulas, which depend on preceding terms, or
• Explicit formulas, which provide a direct computation method for any term.

Sequences are fundamental in mathematics, helping us model everything from simple patterns to complex natural phenomena. They can illustrate progression, change, and correlation, serving as critical building blocks for more advanced mathematical concepts and problem-solving applications. Sequences can be finite, with a set number of terms, or infinite, continuing indefinitely. Understanding sequences involves recognizing their structure and being able to determine or predict terms using the given rules.