Problem 3
Question
Find the common difference d for each arithmetic sequence. Do not use a calculator. $$3,-2,-7,-12, \dots$$
Step-by-Step Solution
Verified Answer
The common difference \(d\) is \(-5\).
1Step 1: Identify the Sequence Terms
The sequence given in the exercise is \(3, -2, -7, -12, \ldots\). This sequence is an arithmetic sequence.
2Step 2: Understand the Formula for Common Difference
The common difference \(d\) in an arithmetic sequence is the difference between any two consecutive terms. This can be calculated using the formula: \(d = a_{n+1} - a_n\), where \(a_n\) and \(a_{n+1}\) are consecutive terms in the sequence.
3Step 3: Choose Consecutive Terms
To find the common difference, choose two consecutive terms from the sequence. Let's choose the first term, \(3\) (\(a_1\)), and the second term, \(-2\) (\(a_2\)).
4Step 4: Calculate the Common Difference
Using the formula \(d = a_{n+1} - a_n\), substitute \(-2\) for \(a_2\) and \(3\) for \(a_1\): \[d = -2 - 3 = -5\]
5Step 5: Verify Consistency of Common Difference
To ensure that the common difference is consistent throughout the sequence, check another pair of consecutive terms. Consider \(-2\) (\(a_2\)) and \(-7\) (\(a_3\)). Apply the formula: \[d = -7 - (-2) = -7 + 2 = -5\] Since the calculated common difference is \(-5\) for both pairs, it confirms consistency.
Key Concepts
Common DifferenceSequence TermsArithmetic Sequence FormulaConsecutive Terms
Common Difference
In an arithmetic sequence, the common difference is a vital feature that determines how each term in the sequence progresses from the one before. It's the constant amount you add (or subtract, in the case of negative numbers) to get from one term to the next. This value is what makes an arithmetic sequence linear. You can easily find it by subtracting any term from the term that follows it.
To understand it simply: imagine walking a straight path and counting your steps. The number of steps you take to reach the next point is the common difference. Once you establish this number, it remains constant throughout the path, just like it does in an arithmetic sequence.
To understand it simply: imagine walking a straight path and counting your steps. The number of steps you take to reach the next point is the common difference. Once you establish this number, it remains constant throughout the path, just like it does in an arithmetic sequence.
Sequence Terms
Sequence terms are the individual elements in a sequence. In arithmetic sequences, these terms are connected by a constant known as the common difference. Each term has a specific position, known as its index or order.
For example, in the sequence given in the exercise, which is represented as \(3, -2, -7, -12, \ldots\), each number is a term of the sequence. The sequence starts with the first term, often denoted by \(a_1\), and continues indefinitely with each term calculated by adding the common difference to the previous term.
For example, in the sequence given in the exercise, which is represented as \(3, -2, -7, -12, \ldots\), each number is a term of the sequence. The sequence starts with the first term, often denoted by \(a_1\), and continues indefinitely with each term calculated by adding the common difference to the previous term.
- First term (\(a_1\)): 3
- Second term (\(a_2\)): -2
- Third term (\(a_3\)): -7
Arithmetic Sequence Formula
The arithmetic sequence formula is a mathematical rule used to determine any term in an arithmetic sequence. It is essential for finding unknown terms or understanding the relationship between terms in a sequence. The formula is expressed as:
\[ a_n = a_1 + (n-1) \, d \]
Here, \(a_n\) is the \(n\)th term you want to find, \(a_1\) is the first term, and \(d\) stands for the common difference. \(n-1\) represents the number of steps you take from the first term to reach \(a_n\).
Using the example sequence, if you wanted to find the 4th term \(a_4\), you would plug the first term \(a_1 = 3\) and the common difference \(d = -5\) into the formula, resulting in:
\[ a_4 = 3 + (4-1)(-5) = 3 - 15 = -12 \]
This illustrates how you can predictably and consistently determine each term of the sequence.
\[ a_n = a_1 + (n-1) \, d \]
Here, \(a_n\) is the \(n\)th term you want to find, \(a_1\) is the first term, and \(d\) stands for the common difference. \(n-1\) represents the number of steps you take from the first term to reach \(a_n\).
Using the example sequence, if you wanted to find the 4th term \(a_4\), you would plug the first term \(a_1 = 3\) and the common difference \(d = -5\) into the formula, resulting in:
\[ a_4 = 3 + (4-1)(-5) = 3 - 15 = -12 \]
This illustrates how you can predictably and consistently determine each term of the sequence.
Consecutive Terms
Consecutive terms are terms that appear one after the other in a sequence. In arithmetic sequences, consecutive terms are connected by the common difference. Understanding consecutive terms is key to recognizing the pattern that defines the sequence.
Consider the sequence again: \(3, -2, -7, -12, \ldots\). Here, between each pair of consecutive terms, there's a common difference of \(-5\). Each term follows the one before it by subtracting 5.
Consider the sequence again: \(3, -2, -7, -12, \ldots\). Here, between each pair of consecutive terms, there's a common difference of \(-5\). Each term follows the one before it by subtracting 5.
- From 3 to -2: \(-5\)
- From -2 to -7: \(-5\)
- From -7 to -12: \(-5\)
Other exercises in this chapter
Problem 2
Write the terms of the geometric sequence that satisfies the given conditions. $$a_{1}=-\frac{3}{4}, r=\frac{2}{3}, n=4$$
View solution Problem 2
Write the first five terms of each sequence. Do not use a calculator. $$a_{n}=6 n-3$$
View solution Problem 3
Evaluate the following. In Exercises 17 and \(18,\) express the answer in terms of \(n .\) Do not use a calculator. $$\frac{7 !}{3 ! 4 !}$$
View solution Problem 3
For which positive integers is the statement 2^{n}>2 n not true?
View solution