Problem 11 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify the Initial Term

The sequence is defined recursively and begins with an initial term. Given that \(a_1 = 3\), this is our starting point.

Step 2: Use the Recursive Formula for the Second Term

Plug \(a_1 = 3\) into the recursive formula \(a_n = 2(a_{n-1} - 2)\) to find \(a_2\).\[a_2 = 2(a_1 - 2) = 2(3 - 2) = 2 \times 1 = 2\]Thus, \(a_2 = 2\).

Step 3: Calculate the Third Term

Use the value of \(a_2 = 2\) to find \(a_3\).\[a_3 = 2(a_2 - 2) = 2(2 - 2) = 2 \times 0 = 0\]Therefore, \(a_3 = 0\).

Step 4: Determine the Fourth Term

Find \(a_4\) using \(a_3 = 0\).\[a_4 = 2(a_3 - 2) = 2(0 - 2) = 2 \times (-2) = -4\]Hence, \(a_4 = -4\).

Step 5: Compute the Fifth Term

Using \(a_4 = -4\), find \(a_5\).\[a_5 = 2(a_4 - 2) = 2(-4 - 2) = 2 \times (-6) = -12\]So, \(a_5 = -12\).

Key Concepts

Arithmetic Sequences

An arithmetic sequence is a list of numbers in which the difference between consecutive terms is constant. This constant difference is known as the "common difference." In simpler terms, you add the same value each time you move from one term to the next. Arithmetic sequences are deeply valued in mathematics
as they make complex problems easier to solve due to their linear nature.
Some characteristics of arithmetic sequences include:

• The general formula for the n-th term of an arithmetic sequence, given the first term \(a_1\) and the common difference \(d\), is \(a_n = a_1 + (n-1) \times d\).
• They can be finite or infinite.
• They appear in numerous real-life settings, like calculating interest rates and predicting future events based on linear progressions.

While the sequence in the exercise is defined recursively, understanding arithmetic sequences helps grasp different ways terms relate to one another.

Recursive Formulas

Recursive formulas provide a method to define a sequence using its previous term(s). Instead of expressing terms with a direct equation as in arithmetic sequences, recursive formulas connect each term to its predecessor. This approach often builds sequences step-by-step, starting from a given initial term.In the given problem, the formula is \(a_n = 2(a_{n-1} - 2)\). Here's how it works:

• You calculate each new term based directly on the term before it.
• To find \(a_2\), you use \(a_1\), to find \(a_3\), use \(a_2\), and so forth.
• It often involves a base case or initial condition—here, \(a_1 = 3\), giving you a starting point.

Recursive methods can model natural processes and growth patterns in real-world scenarios. While slightly different in concept from arithmetic sequences, recursive formulas are powerful tools for calculations and predictions over successive terms.

Sequence Terms

Sequence terms are the individual elements or numbers in a sequence. Each term is identified by its position: first, second, third, and so on. In computations or analysis, terms offer targeted insights into the underlying pattern of a sequence.To analyze sequence terms, consider:

• The formula generating them—in our case, recursive.
• Their relationship with prior or subsequent terms.
• The context in which they appear, such as number series, function outputs, or algorithm steps.

In our exercise, the terms \(a_1, a_2, a_3, a_4,\) and \(a_5\) were found incrementally:
- \(a_1 = 3\): The initial term, provided as part of the sequence definition.- \(a_2 = 2\): Calculated using \(a_1\).- \(a_3 = 0\): Found by substituting \(a_2\) in the recursive formula.- \(a_4 = -4\): Derived by applying the same formula to \(a_3\).- \(a_5 = -12\): The fifth term, emerging from substituting \(a_4\).Recognizing how each term builds upon the previous highlights the logical flow and relationship within sequences, serving both theoretical inquiry and practical applications.