Problem 46 - Algebra 2

Question

Find the first five terms of each sequence. $$ a_{1}=7, a_{n+1}=a_{n}-2 $$

Step-by-Step Solution

Answer

The first five terms are 7, 5, 3, 1, -1.

Step 1: Understand the Given Information

The sequence is defined recursively with the first term given as \( a_1 = 7 \). Each subsequent term is formed by subtracting 2 from the previous term as given by the formula \( a_{n+1} = a_n - 2 \). We need to find the first five terms of this sequence.

Step 2: Calculate the Second Term

Using the relationship \( a_{n+1} = a_n - 2 \) and the given \( a_1 = 7 \), we find the second term: \[ a_2 = a_1 - 2 = 7 - 2 = 5 \]

Step 3: Calculate the Third Term

Continuing with the recursive relationship:\[ a_3 = a_2 - 2 = 5 - 2 = 3 \]

Step 4: Calculate the Fourth Term

Using the rule \( a_{n+1} = a_n - 2 \):\[ a_4 = a_3 - 2 = 3 - 2 = 1 \]

Step 5: Calculate the Fifth Term

Apply the recursive formula again:\[ a_5 = a_4 - 2 = 1 - 2 = -1 \]

Step 6: Summarize the First Five Terms

The first five terms of the sequence are: 7, 5, 3, 1, -1.

Key Concepts

Arithmetic Sequences

Arithmetic sequences are a type of sequence in which the difference between consecutive terms is constant. This difference is known as the common difference. In the context of numerical sequences, arithmetic sequences are among the simplest and are widely studied due to their straightforward properties. For an arithmetic sequence, the general form is: \( a_n = a_1 + (n - 1) \cdot d \).
Here:

• \( a_1 \) represents the first term of the sequence.
• \( d \) is the common difference.
• \( n \) is the term number.

In our original exercise, we begin with \( a_1 = 7 \). Each subsequent term decreases by 2, which means the common difference \( d \) is \(-2\). Understanding this concept is crucial as it sets the foundation for predicting any term in the sequence once you know a single term and the common difference.

Sequence Terms

Each number in a sequence is referred to as a 'term'. When working through a sequence, you calculate its terms in a specific order, typically starting from a given first term. Knowing how to find these terms is essential to understanding the pattern and behavior of sequences.In our exercise, the first term \( a_1 \) is given as 7. We use the recursive rule \( a_{n+1} = a_n - 2 \) to calculate the following terms: 5, 3, 1, and finally -1.
This progression demonstrates a clear pattern of subtraction, characterizing the sequence and making it predictable beyond just its initial terms.Understanding individual terms and the rules that generate them will aid greatly in identifying the nature of a sequence, whether it be finite, infinite, or adhere to some form of convergence or divergence.

Recurrence Relation

A recurrence relation is a way of defining the terms of a sequence using the previous terms. They are especially useful for sequences where it is convenient or necessary to compute terms step by step, rather than jumping directly to a formula. In the provided exercise, the sequence is defined using the recurrence relation \( a_{n+1} = a_n - 2 \). This mathematical statement means that each term is derived by subtracting 2 from the previous term.

• \( a_1 \) is known as the initial condition or base case since it starts the sequence.
• \( a_n \) is the current term, and \( a_{n+1} \) is the next term.

Recurrence relations can be seen in various forms and complexity, from simple ones like the one in our example to more complex relations seen in advanced mathematics, such as Fibonacci sequences. Understanding recurrence relations allows mathematicians and students alike to work with complex sequences efficiently, finding patterns and solutions where direct formulas might not be obvious.