Problem 9

Question

Write the first five terms of each sequence. Do not use a calculator. $$a_{n}=(-1)^{n}(2 n)$$

Step-by-Step Solution

Verified

Answer

The first five terms are: -2, 4, -6, 8, -10.

1Step 1: Understand the General Formula

The formula given is $a_{n}=(-1)^{n}(2n)$. This formula is composed of two parts: $(-1)^{n}$ and $2n$. The $(-1)^{n}$ part will determine the sign of the term (i.e., positive if $n$ is even and negative if $n$ is odd), while $2n$ will give the magnitude of the term, multiplying $n$ by 2.

2Step 2: Calculate the First Term

To find the first term $a_1$, substitute $n=1$ into the formula: \[a_1 = (-1)^1(2 \times 1) = -2\].

3Step 3: Calculate the Second Term

Substitute $n=2$ into the formula to find the second term $a_2$: \[a_2 = (-1)^2(2 \times 2) = 4\].

4Step 4: Calculate the Third Term

Substitute $n=3$ into the formula to find the third term $a_3$: \[a_3 = (-1)^3(2 \times 3) = -6\].

5Step 5: Calculate the Fourth Term

Substitute $n=4$ into the formula to find the fourth term $a_4$: \[a_4 = (-1)^4(2 \times 4) = 8\].

6Step 6: Calculate the Fifth Term

Substitute $n=5$ into the formula to find the fifth term $a_5$: \[a_5 = (-1)^5(2 \times 5) = -10\].

Key Concepts

Sequence FormulaAlgebraic ExpressionsSign of Terms

Sequence Formula

When working with sequences, a sequence formula is crucial for determining the terms of the sequence. In this exercise, the provided sequence formula is $a_{n}=(-1)^{n}(2n)$. This formula serves as a blueprint for generating each term of the sequence based on its position $n$.

Understanding the components of the sequence formula allows us to predict the value of each term easily. The formula is split into two parts: $(-1)^{n}$ and $2n$. The

$(-1)^{n}$ part determines the sign: negative when $n$ is odd, and positive when $n$ is even.
$2n$ part represents the magnitude, multiplying $n$ by 2 to determine the absolute value of the term.

Thus, the formula organizes the sequence systematically, alternating the signs based on $n$, while increasing in size by a consistent pattern, determined by the simple multiplication of 2 with $n$. Understanding how sequence formulas function and their components is essential in expanding or finding terms in arithmetic sequences.

Algebraic Expressions

An algebraic expression is a mathematical phrase involving numbers, variables, and operational symbols. In the sequence formula $a_{n}=(-1)^{n}(2n)$, you can see an algebraic expression if you look closely. It combines constants and variables to define the sequence in mathematical terms.

Here's a breakdown:

$(-1)^{n}$ is an exponential expression concerning the variable $n$ which impacts the sign of the sequence.
$2n$ creates a linear expression where $n$ is multiplied by 2, establishing the size of the sequence term.

Algebraic expressions are powerful because they generalize patterns and make calculations easier. They provide the foundation for understanding sequences and where each component fits within the overall computation. By decomposing the sequence formula into its parts, you understand not only the individual components but how each contributes to forming the entire sequence structure.

Sign of Terms

In sequences like $a_{n}=(-1)^{n}(2n)$, the sign of each term changes systematically, providing insight into arithmetic sequences that feature alternating patterns. This alternating nature is controlled by the $(-1)^{n}$ component of the sequence formula.

With every increment in $n$:

If $n$ is even (such as 2, 4, 6), then $(-1)^{n}$ equals 1, and the term will be positive.
If $n$ is odd (such as 1, 3, 5), then $(-1)^{n}$ equals -1, making the term negative.

This systematic alternation exemplifies how mathematical expressions control and describe changes in sequences. They help predict the behavior and flow of a sequence even before calculations are made.

Understanding these patterns enhances the grasp of sequences, especially when tackling alternating arithmetic sequences, allowing for easier computation and visualization of the sequence's characteristics.

Problem 9

Problem 10

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Problem 9

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Problem 10

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