Problem 55 - Algebra 2

Question

Use each recursive formula to write an explicit formula for the sequence. $$ a_{1}=1, a_{n}=a_{n-1}+4 $$

Step-by-Step Solution

Answer

The explicit formula for the sequence \(a_{1}=1, a_{n}=a_{n-1}+4\) is \(a_n=4n-3\).

Step 1: Understand the Recursive Rule

The recursive rule we have is \(a_{n}=a_{n-1}+4\). This means each term is just the previous term plus 4. For example, to get the second term, you add 4 to the first term.

Step 2: Write Out Initial Terms

By using the recursive rule, we can generate the sequence up to a few terms: \(a_1=1\), \(a_2=1+4=5\), \(a_3=5+4=9\), \(a_4=9+4=13\), and so on.

Step 3: Identify the Pattern for an Explicit Formula

Looking at these terms, we can see a pattern. Each term can be expressed as: \(a_n=4n-3\). This should be checked against the initial terms for validity.

Step 4: Validate the Explicit Formula

We check this formula with our initial terms: when \(n=1\), \(a_1=4*1-3=1\); when \(n=2\), \(a_2=4*2-3=5\); when \(n=3\), \(a_3=4*3-3=9\), etc. Our explicit formula works.

Step 5: Write the Final Answer

So the explicit formula for the sequence \(a_{1}=1, a_{n}=a_{n-1}+4\) is \(a_n=4n-3\).

Key Concepts

Recursive Sequence

Sequences are an essential part of mathematics. A _recursive sequence_ is a sequence in which each term is defined based on one or more previous terms. The sequence given in the exercise begins with the term \(a_1 = 1\). Each subsequent term is calculated using the recursive rule \(a_n = a_{n-1} + 4\). This means each term in the sequence depends on the one directly before it.

Understanding recursive sequences is crucial because it allows us to build sequences incrementally from a starting point. They are like building blocks where each element is a step forward from the previous one. Such sequences are typically easy to compute but may sometimes obscure the relationship between the sequence's position and its value.

Recursive Formula

A _recursive formula_ provides a method for finding the next term(s) in a sequence using the terms that came before it. This is evident in our exercise with the formula \(a_n = a_{n-1} + 4\). The formula tells us how to derive the current term from the last one by simply adding 4.

Recursive formulas are powerful because they can model many natural phenomena and processes. They offer simplicity and clarity when describing patterns. However, recognizing an explicit formula from a recursive one is valuable because it highlights a direct relationship between a term's position and its value, enabling quicker calculations.

Arithmetic Sequence

An _arithmetic sequence_ is a sequence of numbers in which the difference between any two consecutive terms is constant. This sequence is often recognized by its common difference. In our exercise, the sequence starts with \(a_1 = 1\) and continues with \(a_2 = 5\), \(a_3 = 9\), etc., each time increasing by 4.

The explicit formula derived from this sequence is \(a_n = 4n - 3\). Each number can be calculated by plugging in its position \(n\) into this formula, which eliminates the need to calculate all preceding values. Understanding arithmetic sequences and their explicit formulas is essential as they are simple yet widely applicable in various mathematical and real-world situations. The explicit formula offers a straightforward way to find any term in the sequence directly, saving time and effort when dealing with large or unknown terms.