Problem 10 - Contemporary Precalculus

Question

Find the first five terms of the sequence \(\left\\{a_{n}\right\\}\). $$a_{n}=5-(.1)^{n}$$

Step-by-Step Solution

Answer

Answer: The first five terms of the sequence are $$a_1 = 4.9$$, $$a_2 = 4.99$$, $$a_3 = 4.999$$, $$a_4 = 4.9999$$, and $$a_5 = 4.99999$$.

Step 1: Identify the first term of the sequence#a_n for n=1

To find the first term of the sequence, plug n = 1 into the term expression: $$a_1 = 5 - (.1)^1$$.

Step 2: Compute the first term#a_1

Calculate the term by solving the exponent and performing the subtraction: $$a_1 = 5 - 0.1 = 4.9$$.

Step 3: Identify the second term of the sequence#a_n for n=2

To find the second term of the sequence, plug n = 2 into the term expression: $$a_2 = 5 - (.1)^2$$.

Step 4: Compute the second term#a_2

Calculate the term by solving the exponent and performing the subtraction: $$a_2 = 5 - 0.01 = 4.99$$.

Step 5: Identify the third term of the sequence#a_n for n=3

To find the third term of the sequence, plug n = 3 into the term expression: $$a_3 = 5 - (.1)^3$$.

Step 6: Compute the third term#a_3

Calculate the term by solving the exponent and performing the subtraction: $$a_3 = 5 - 0.001 = 4.999$$.

Step 7: Identify the fourth term of the sequence#a_n for n=4

To find the fourth term of the sequence, plug n = 4 into the term expression: $$a_4 = 5 - (.1)^4$$.

Step 8: Compute the fourth term#a_4

Calculate the term by solving the exponent and performing the subtraction: $$a_4 = 5 - 0.0001 = 4.9999$$.

Step 9: Identify the fifth term of the sequence#a_n for n=5

To find the fifth term of the sequence, plug n = 5 into the term expression: $$a_5 = 5 - (.1)^5$$.

Step 10: Compute the fifth term#a_5

Calculate the term by solving the exponent and performing the subtraction: $$a_5 = 5 - 0.00001 = 4.99999$$.

Step 11: List the first five terms of the sequence

The first five terms of the sequence are: $$a_1 = 4.9$$, $$a_2 = 4.99$$, $$a_3 = 4.999$$, $$a_4 = 4.9999$$, and $$a_5 = 4.99999$$.

Key Concepts

Precalculus: Understanding Sequences

A sequence in mathematics refers to an ordered list of numbers, each of which is called a term. In precalculus, understanding sequences is fundamental for exploring how numbers can follow specific patterns. When tackling sequences, it's crucial to know how each term relates to its position, defined by the variable \(n\) in the sequence formula. For example, in our original problem, we are asked to examine the sequence \(\{a_n\}\) given by the formula \(a_n = 5 - (0.1)^n\). Here, you notice that \(a_n\) varies as \(n\) changes, demonstrating part of the broader field of sequences and series in mathematics. In this context, addressing sequences enhances our grasp of mathematical behavior paving the way for higher-level algebraic and calculus concepts.

Sequence and Series: Arithmetic Sequences

Arithmetic sequences are a type of sequence where each term is created by adding a fixed number, known as the 'common difference', to the previous term. Though the sequence seen in the exercise is not arithmetic (since it follows an exponential pattern), understanding arithmetic sequences helps to establish a basic understanding of regularity in sequences. They are characterized by their uniform increment or decrement. For example, the sequence \(2, 4, 6, 8, \ldots\) has a constant difference of 2. Contrast this with the exponential nature of sequences such as \(a_n = 5 - (0.1)^n\), which change at a rate that depends on the power of 10. Recognizing the difference between arithmetic and non-arithmetic sequences is vital in determining which mathematical approach to apply.

Exponential Decay and Sequence Behavior

Exponential decay describes the process where quantities decrease rapidly at first and slow down over time. In the formula \(a_n = 5 - (0.1)^n\), the term \((0.1)^n\) represents a diminishing component, highlighting this concept. Each term in our sequence becomes closer to the constant value 5 as \(n\) increases. Here's the beautiful part of exponential decay:

• It never quite reaches the limit – 5 in this case.
• It mirrors many real-world scenarios, such as radioactive decay or the cooling of an object.

Such sequences illustrate how exponential functions can describe rapid initial changes that flatten out over time. Understanding exponential decay provides powerful insights into both mathematical and practical phenomena, emphasizing how numbers can model real-world processes.