Problem 12 - Beginning and Intermediate Algebra

Question

Write out the first five terms of each sequence. $$a_{n}=(-1)^{n} \cdot(n+1)$$

Step-by-Step Solution

Answer

The first five terms of the sequence are: -2, 3, -4, 5, -6.

Step 1: Understand the formula

The given formula for the sequence is: \(a_{n}=(-1)^{n}(n+1)\) To find the terms of the sequence, we'll plug in the values of 'n' from 1 to 5 into the formula.

Step 2: Find the first term (n=1)

For n=1, the formula becomes: \(a_{1}=(-1)^{1}(1+1)\) Evaluate the expression: \(a_{1}=(-1)(2)\) \(a_{1}=-2\)

Step 3: Find the second term (n=2)

For n=2, the formula becomes: \(a_{2}=(-1)^{2}(2+1)\) Evaluate the expression: \(a_{2}=(1)(3)\) \(a_{2}=3\)

Step 4: Find the third term (n=3)

For n=3, the formula becomes: \(a_{3}=(-1)^{3}(3+1)\) Evaluate the expression: \(a_{3}=(-1)(4)\) \(a_{3}=-4\)

Step 5: Find the fourth term (n=4)

For n=4, the formula becomes: \(a_{4}=(-1)^{4}(4+1)\) Evaluate the expression: \(a_{4}=(1)(5)\) \(a_{4}=5\)

Step 6: Find the fifth term (n=5)

For n=5, the formula becomes: \(a_{5}=(-1)^{5}(5+1)\) Evaluate the expression: \(a_{5}=(-1)(6)\) \(a_{5}=-6\)

Step 7: Write out the first five terms

The first five terms of the given sequence are as follows: -2, 3, -4, 5, -6

Key Concepts

Algebraic Expressions

Algebraic expressions form the backbone of many mathematical problems and solutions. They are combinations of numbers, variables, and arithmetic operations that model relationships and processes. Understanding algebraic expressions is essential for making sense of mathematical sequences.
Algebraic expressions can be simple like \( x + 2 \) or complex like \( (-1)^n (n+1) \). The expression \( (-1)^n (n+1) \) consists of:

• A base \((-1)^n\), responsible for the alternating sign.
• A multiplication operator ensuring the combined effect of both parts.
• The term \((n+1)\) representing the increasing numeric component as \(n\) advances.

Breaking down expressions into these components helps in understanding their role when plugged into sequence formulas. Each component plays a specific role in defining the output values.

Sequence Terms

In any sequence, terms are the individual elements that make up the sequence. They are ordered and can be finite or infinite. For our example, the sequence is determined by the formula \(a_n = (-1)^n (n+1)\).
The sequence terms are formed by evaluating the formula for consecutive values of \(n\), which in our exercise, are \(n = 1, 2, 3, 4, 5\). Substituting these values gives each term of the sequence:

• For \(n=1\), the term is \(-2\).
• For \(n=2\), the term is \(3\).
• For \(n=3\), the term is \(-4\).
• For \(n=4\), the term is \(5\).
• For \(n=5\), the term is \(-6\).

Writing terms in this systematic way ensures clarity and accuracy in representing the sequence.

Alternating Sequence

An alternating sequence is one where the signs of the terms switch between positive and negative in a regular pattern. This type of sequence is achieved through a specific component in the formula: \((-1)^n\).
The expression \((-1)^n\) determines the sign of the sequence terms. Here's how it works:

• When \(n\) is an odd number, \((-1)^n = -1\), making the term negative.
• When \(n\) is an even number, \((-1)^n = 1\), making the term positive.

This alternating pattern of signs aids in generating sequence types that switch between representing increases and decreases, oscillating around zero or any other number.
Recognizing alternating sequences helps in predicting behavior over larger sequences or complex mathematical models.

Evaluation of Sequences

Evaluating a sequence involves calculating its terms using the provided formula. For our example, the key is substituting integer values of \(n\) to obtain the terms. Here's a step-by-step approach for evaluating each term:1. Identify the value of \(n\) for the desired term.2. Substitute \(n\) into the sequence formula, \(a_n = (-1)^n (n+1)\).3. Calculate \((-1)^n\) to decide the sign of the term.4. Compute \((n+1)\), which contributes to the magnitude of the term.5. Multiply the results to find the sequence term.Applying these steps systematically ensures each term is derived accurately. By practicing these steps, you build a strong foundation in handling more complicated sequences, enhancing your ability to analyze mathematical progressions.