Problem 14 - College Algebra with Corequisite Support

Question

For the following exercises, write the first four terms of the sequence. $$ a_{n}=(-10)^{n}+1 $$

Step-by-Step Solution

Answer

The first four terms are -9, 101, -999, and 10001.

Step 1: Identify the Terms to Find

We need the first four terms of the sequence. These terms are: \(a_1, a_2, a_3,\) and \(a_4\).

Step 2: Calculate the First Term

Substitute \(n = 1\) into the formula: \[a_1 = (-10)^1 + 1 = -10 + 1 = -9\]

Step 3: Calculate the Second Term

Substitute \(n = 2\) into the formula: \[a_2 = (-10)^2 + 1 = 100 + 1 = 101\]

Step 4: Calculate the Third Term

Substitute \(n = 3\) into the formula: \[a_3 = (-10)^3 + 1 = -1000 + 1 = -999\]

Step 5: Calculate the Fourth Term

Substitute \(n = 4\) into the formula: \[a_4 = (-10)^4 + 1 = 10000 + 1 = 10001\]

Key Concepts

Arithmetic Sequences

Arithmetic sequences are all about a simple addition or subtraction applied regularly. Consider a perfectly organized sequence of numbers where each term after the first is formed by adding a constant to the previous term. This constant is known as the "common difference". For example, imagine the sequence: 2, 5, 8, 11, ... Here, you can see each number after 2 is created by adding 3, which makes 3 the common difference.

Key features of arithmetic sequences include:

• The formulas are straightforward. The nth term of an arithmetic sequence: \[ a_n = a_1 + (n-1)d \]where \(a_1\) is the first term, and \(d\) is the common difference.
• They form a linear pattern when graphed.
• Predictable and easy to manage as they slide smoothly from one number to the next.

Geometric Sequences

Now, let's explore geometric sequences, which expand exponentially by multiplying terms by a fixed number. This fixed multiplier is known as the "common ratio". Picture the sequence: 3, 6, 12, 24, ... where each number is the product of its predecessor rolled out by the common ratio of 2.

Understanding geometric sequences becomes clearer when recognizing:

• The formula for the nth term of a geometric sequence is: \[ a_n = a_1 \, r^{(n-1)} \]where \(a_1\) is the first term, and \(r\) is the common ratio.
• They create exponential curves on a graph.
• Their values can increase or decrease significantly depending on whether the common ratio is greater than or less than one.

Sequence Terms Calculation

Calculating sequence terms involves substituting values into a predefined formula. This step-by-step substitution can determine any term in a sequence. Let's break it down with the given exercise:

1. Identify what is needed: typically, the formula for a sequence and the specific terms you need.2. Substitute the appropriate index number into the sequence formula.3. Perform the arithmetic operations to find the result.
For this exercise where the sequence is defined by \(a_{n} = (-10)^{n}+1\):

• The first term \(a_1\) is computed simply by letting \(n = 1\): \((-10)^1 + 1 = -9\).
• Repeat with \(n = 2, 3,\) and \(4\) to discover the subsequent terms like 101, -999, and 10001.

This method applies not just to this sequence but widely across both arithmetic and geometric sequences.

Algebraic Expressions

Algebraic expressions form the backbone of sequences. In essence, they are mathematical phrases using numbers, variables, and operation symbols to represent real-world quantities.

In sequences, whether arithmetic or geometric, an algebraic expression serves as the defining formula from which sequence terms are generated. Take the expression \((-10)^n + 1\) from the exercise.

In this case, each term is determined by computing the power of \(-10\) to the n-th degree:

• Operates as the main computational component, determining how terms switch between positive and negative or large and larger values.
• Incorporates constants like +1 that modify the overall value of the terms.
• Flexible, allowing adjustments to explore various sequence patterns by changing coefficients or powers.

Draw a clear distinction: while the expression outputs a sequence's terms, it should not be confused with just a simple arithmetic or geometric pattern. It might sometimes create unique progressions not fitting those categories.