Problem 13 - Algebra and Trigonometry

Question

\(13-18\) Find the first five terms of the given recursively defined sequence. $$ a_{n}=2\left(a_{n-1}-2\right) \quad \text { and } \quad a_{1}=3 $$

Step-by-Step Solution

Answer

The first five terms are 3, 2, 0, -4, -12.

Step 1: Calculate the Second Term

To find the second term of the sequence, substitute \(n = 2\) into the recursive formula: \(a_2 = 2(a_1 - 2)\). Given that \(a_1 = 3\), substitute to find \(a_2 = 2(3 - 2) = 2 \times 1 = 2\).

Step 2: Calculate the Third Term

Now, find the third term by substituting \(n = 3\) into the recursive formula: \(a_3 = 2(a_2 - 2)\). Since \(a_2 = 2\), substitute to find \(a_3 = 2(2 - 2) = 2 \times 0 = 0\).

Step 3: Calculate the Fourth Term

Find the fourth term by substituting \(n = 4\) into the recursive formula: \(a_4 = 2(a_3 - 2)\). Since \(a_3 = 0\), substitute to find \(a_4 = 2(0 - 2) = 2 \times (-2) = -4\).

Step 4: Calculate the Fifth Term

Find the fifth term by substituting \(n = 5\) into the recursive formula: \(a_5 = 2(a_4 - 2)\). Given \(a_4 = -4\), substitute to find \(a_5 = 2(-4 - 2) = 2 \times (-6) = -12\).

Step 5: Summarize the First Five Terms

The calculated terms of the sequence are: \(a_1 = 3\), \(a_2 = 2\), \(a_3 = 0\), \(a_4 = -4\), and \(a_5 = -12\).

Key Concepts

Sequence Terms

In mathematics, a sequence is a list of numbers arranged in a specific order. Each number in the list is called a term. Understanding sequence terms is essential because they are the building blocks of a sequence. Sequence terms are usually denoted as \(a_n\), where \(n\) represents the term's position in the overall sequence.

Let's dive deeper into the example from the original exercise to see this in action.

• The first term, \(a_1\), is given as \(3\).
• The second term, \(a_2\), is calculated using the recursive formula, which we will discuss in the next section.
• Each term depends on the preceding term, showing how sequence terms relate to one another.

Once you get a hold of the first few terms, you can start to see a pattern emerge in the sequence, which further helps in understanding complex problems.

Recursive Formula

A recursive formula is a rule for determining the terms of a sequence using the preceding terms. This is the main difference from a direct, or explicit, formula.

• In simple terms, a recursive formula helps find the next term if you know the previous term.
• The formula given in the problem is \(a_n = 2(a_{n-1} - 2)\).
• This formula dictates that each term is twice the value of the previous term minus two.

With recursive sequences, all subsequent terms rely on the initial term(s) and their calculations to continue the sequence. Recursive formulas can be very useful for sequences that are quite complex because they break down the sequence into smaller, more manageable pieces.

Sequence Calculation

Sequence calculation involves the process of finding such terms using the recursive formula, starting from the initial term. Here, we will outline how each term was calculated in the exercise.

• First Term: This is given as \(a_1 = 3\). It's the starting point of the sequence.
• Second Term: Using \(a_2 = 2(a_1 - 2)\), and substituting \(a_1 = 3\), we find \(a_2 = 2(3 - 2) = 2 \times 1 = 2\).
• Third Term: From \(a_3 = 2(a_2 - 2)\) and \(a_2 = 2\), it follows that \(a_3 = 2(2 - 2) = 2 \times 0 = 0\).
• Fourth Term: Using \(a_4 = 2(a_3 - 2)\), and knowing \(a_3 = 0\), we calculate \(a_4 = 2(0 - 2) = -4\).
• Fifth Term: Plug \(a_4 = -4\) into \(a_5 = 2(a_4 - 2)\), giving \(a_5 = 2(-4 - 2) = -12\).

Calculating sequence terms using a recursive formula demands careful attention to each calculation step. With practice, this method becomes a powerful tool to handle more complex series of numbers.