Problem 12 Find the first term and the comm... [FREE SOLUTION]

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Chapter 14: Problem 12

Find the first term and the common difference. $$ -8,-5,-2,1, \ldots $$

Short Answer

Expert verified

First term: -8, Common difference: 3

Step by step solution

Identify the given sequence

The given sequence is $-8, -5, -2, 1, \ldots$.

- Identify the first term

The first term ($a_1$) is always the initial number in the sequence. Therefore, $a_1 = -8$.

- Find the common difference

The common difference ($d$) in an arithmetic sequence is found by subtracting the first term from the second term. \[-5 - (-8) = -5 + 8 = 3\] So, the common difference is $d = 3$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

First Term

The first term in an arithmetic sequence is simply the initial number in the list of numbers. This value is denoted as $a_1$. Identifying the first term is crucial because it sets the starting point for the entire sequence.
In the given example, the sequence starts with $-8$. Therefore, the first term $a_1$ is -8. This step is quite straightforward: just look at the very first number!

Common Difference

The common difference in an arithmetic sequence is the amount by which each term increases or decreases as you move from one term to the next. It is represented by the letter $d$. To calculate the common difference, subtract the first term from the second term or any subsequent term from the previous one.
In our example, the first term is $a_1 = -8$ and the second term is $a_2 = -5$. Using the formula:

Common difference, $(d) = a_2 - a_1$
\[d = -5 - (-8)\]
\[d = -5 + 8\]
\[d = 3\]

So, the common difference ($d$) is 3.

Sequence Identification

Understanding whether a list of numbers forms an arithmetic sequence involves checking if there is a consistent pattern. Specifically, you look to see if the difference between consecutive terms is constant.
For the sequence $ -8, -5, -2, 1, \ldots $, we start by finding the differences:

$-5 - (-8) = 3$
$-2 - (-5) = 3$
$1 - (-2) = 3$

Since the difference $d = 3$ is consistent between each pair of consecutive terms, we can confidently say that this is an arithmetic sequence. Consistency in the common difference is the key characteristic that helps in identifying this kind of sequence.

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91影视

Find the first term and the common difference. $$ -8,-5,-2,1, \ldots $$

Short Answer

Step by step solution

Identify the given sequence

- Identify the first term

- Find the common difference

Key Concepts

First Term

Common Difference

Sequence Identification

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