Problem 26 - Contemporary Precalculus

Question

Find the first five terms of the recursively defined sequence. $$a_{1}=0 \text { and } a_{n}=3 a_{n-1}-2 \text { for } n \geq 2$$

Step-by-Step Solution

Answer

Answer: The first five terms of the sequence are \(a_1 = 0\), \(a_2 = -2\), \(a_3 = -8\), \(a_4 = -26\), and \(a_5 = -80\).

Step 1: 1. Identify the given first term#a_1\(

The first term of the sequence is given as \)a_1=0$.

Step 2: 2. Calculate \(a_2\) by applying the given formula

Now, use the formula to find the second term of the sequence: $$a_2=3a_{2-1}-2=3a_1-2=3(0)-2=-2$$ So, \(a_2 = -2\).

Step 3: 3. Calculate \(a_3\) by applying the given formula

Next, find the third term of the sequence: $$a_3=3a_{3-1}-2=3a_2-2=3(-2)-2=-6-2=-8$$ So, \(a_3 = -8\).

Step 4: 4. Calculate \(a_4\) by applying the given formula

Now, find the fourth term of the sequence: $$a_4=3a_{4-1}-2=3a_3-2=3(-8)-2=-24-2=-26$$ So, \(a_4 = -26\).

Step 5: 5. Calculate \(a_5\) by applying the given formula

Finally, find the fifth term of the sequence: $$a_5=3a_{5-1}-2=3a_4-2=3(-26)-2=-78-2=-80$$ So, \(a_5 = -80\).

Step 6: 6. List the first five terms of the sequence

The first five terms of the recursively defined sequence are: $$a_1 = 0, \ a_2 = -2, \ a_3 = -8, \ a_4 = -26, \ a_5 = -80$$

Key Concepts

Sequence and Series

A sequence is a list of numbers or objects in a special order. In precalculus, sequences are important, because they're foundational to understanding more complex mathematical concepts. Sequences can be finite, with a definite number of terms, or infinite, where they go on forever.

In the case of series, it is the sum of the elements of a sequence. If you were to add up all the terms in the sequence, you would get a series. A simple example is the sequence of natural numbers 1, 2, 3, etc., where the series is the sum of these numbers, e.g., 1 + 2 + 3 and so on. The sequence provided in our exercise is a precursor to understanding series since if we were to sum the terms found, we'd be looking at a series formed by this sequence.

A sequence can be thought of a function, where each term corresponds to a unique point in the domain, usually consisting of positive integers. So, understanding sequences is a stepping stone to master functions, a key concept in precalculus and beyond.

Recursive Formula

Recursive formulas define terms of a sequence in relation to previous terms. If you're given a starting point and a rule for finding each successive term based on the terms that came before, you're dealing with a recursive sequence.

The exercise displayed a recursive formula provided as \( a_n=3a_{n-1}-2 \). To find any term in the sequence, you use the formula and plug in the previous term's value. For instance, to find \( a_4 \), we need \( a_3 \). The simplicity of these rules allows us to create complex sequences from simple beginnings.

Working with recursive formulas helps students enhance logical thinking and problem-solving skills, since they compel the thinker to consider the sequence's history to determine its future. As seen in the solution, we used the formula and previous terms to find the values iteratively, showcasing the 'recurrence' in recursive sequences.

Precalculus

Precalculus is a course that prepares students for calculus, the study of change. It covers various topics such as functions, sequences, series, trigonometry, and the beginnings of analysis. Sequences appear frequently in precalculus because they are building blocks for understanding limits, a core concept of calculus.

In precalculus, learners develop skills necessary to handle abstract and complex mathematical concepts, fostering a deep understanding of function behaviors and various types of equations. With recursive formulas, for example, students get a peek into how mathematicians model real-world phenomena with functions that depend on their previous values, much like population growth or financial interest. Seeing how the terms progress over time with a recursive formula can also provide a tangible way to connect the dots between algebraic manipulation and the real-world situations these calculations can represent.

Building a strong foundation in precalculus ensures that students are ready for the challenges of calculus and beyond, setting the groundwork for success in STEM fields.