Problem 9 - Contemporary Precalculus

Question

Find the first five terms of the sequence \(\left\\{a_{n}\right\\}\). $$a_{n}=4+(-.1)^{n}$$

Step-by-Step Solution

Answer

The first five terms of the sequence are $$a_{1} = 3.9, a_{2} = 4.01, a_{3} = 3.999, a_{4} = 4.0001,\text{ and } a_{5} = 3.99999$$.

Step 1: Substitute the value of n=1

Plug in \(n=1\) into the formula: $$a_{1} = 4 + (-0.1)^{1}$$ Calculate \(a_1\): $$a_{1} = 4 - 0.1 = 3.9$$

Step 2: Substitute the value of n=2

Plug in \(n=2\) into the formula: $$a_{2} = 4 + (-0.1)^{2}$$ Calculate \(a_2\): $$a_{2} = 4 + 0.01 = 4.01$$

Step 3: Substitute the value of n=3

Plug in \(n=3\) into the formula: $$a_{3} = 4 + (-0.1)^{3}$$ Calculate \(a_3\): $$a_{3} = 4 - 0.001 = 3.999$$

Step 4: Substitute the value of n=4

Plug in \(n=4\) into the formula: $$a_{4} = 4 + (-0.1)^{4}$$ Calculate \(a_4\): $$a_{4} = 4 + 0.0001 = 4.0001$$

Step 5: Substitute the value of n=5

Plug in \(n=5\) into the formula: $$a_{5} = 4 + (-0.1)^{5}$$ Calculate \(a_5\): $$a_{5} = 4 - 0.00001 = 3.99999$$ The first five terms of the sequence are: $$a_{1} = 3.9, a_{2} = 4.01, a_{3} = 3.999, a_{4} = 4.0001,\text{ and } a_{5} = 3.99999$$

Key Concepts

Sequences in Mathematics

In mathematics, a sequence is an ordered list of objects or numbers that follows a specific pattern or rule. Sequences can be finite, with a limited number of terms, or infinite, continuing indefinitely. One common type is the arithmetic sequence, where the difference between consecutive terms is constant.

For instance, consider the sequence provided in our exercise: \( a_{n} = 4 + (-0.1)^{n} \). This sequence doesn't follow a simple arithmetic pattern, as the subsequent terms rely on the power of -0.1 raised to the value of n. As we can see from the calculated first five terms, the values alternate around the constant number 4 with distinctions becoming negligible as n increases.

Distinguishing between different types of sequences is crucial for understanding their properties and behaviors, for example, determining if they converge or diverge, which leads us to the concept of 'convergence of sequences'.

Geometric Sequences

A geometric sequence, on the other hand, differs from the sequence in our exercise. It is a sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number, known as the 'common ratio'. For example, in a geometric sequence with a first term of 2 and a common ratio of 3, the sequence would be \( 2, 6, 18, 54, \ldots \).

Geometric sequences can show growth or decay, such as in populations, finance, or radioactive substances. It's essential to understand this property since the common ratio determines the sequence's behavior. If the common ratio is greater than 1, the sequence grows; if it's between 0 and 1, it decays; if the ratio is negative, the sequence will alternate signs.

Convergence of Sequences

The convergence of a sequence is a fundamental concept in mathematics, particularly in calculus. A sequence converges if it approaches a specific value, called the limit, as the number of terms goes to infinity. To determine if the sequence from our exercise converges, we analyze its behavior as \( n \) increases.

Considering \( a_{n} = 4 + (-0.1)^{n} \), observe that \( (-0.1)^{n} \) approaches zero as \( n \) becomes large. Thus, \( a_{n} \) approaches 4, making us conclude that this sequence is indeed convergent, and its limit is 4.

Understanding convergence helps in several areas, including integration, series, and differential equations, and sets the stage for more advanced topics such as analysis and complex mathematical modeling in diverse fields.