Problem 43
Question
For the following exercises, write an explicit formula for each arithmetic sequence. $$ a=\\{3,5,7, \ldots\\} $$
Step-by-Step Solution
Verified Answer
The explicit formula is \( a_n = 2n + 1 \).
1Step 1: Identify the First Term
Observe that the first term of the sequence is 3. We will denote the first term as \( a_1 \), so \( a_1 = 3 \).
2Step 2: Calculate the Common Difference
Determine the difference between consecutive terms (second term minus the first term). Here, the second term is 5 and the first term is 3, so the common difference \( d \) is \( 5 - 3 = 2 \).
3Step 3: Write the General Formula for the Arithmetic Sequence
The general formula for the \( n \)-th term of an arithmetic sequence can be written as \( a_n = a_1 + (n-1) \, d \).
4Step 4: Substitute Values into the Formula
Substitute \( a_1 = 3 \) and \( d = 2 \) into the general formula: \( a_n = 3 + (n-1) \, 2 \).
5Step 5: Simplify the Formula
Simplify the expression to arrive at the explicit formula: \( a_n = 3 + 2n - 2 = 2n + 1 \).
Key Concepts
Explicit FormulaCommon DifferenceGeneral Formula
Explicit Formula
In the world of arithmetic sequences, the explicit formula is your key to unlocking any term within the sequence. It's a handy tool that allows you to directly find the value of the nth term without having to go through each term one-by-one from the start. This powerful formula is structured in the form:
The magic of the explicit formula lies in its efficiency. Once you have the first term and the common difference, you can jump directly to any term. For example, looking at the arithmetic sequence \( \{3, 5, 7, \ldots\} \), the explicit formula simplifies to \( a_n = 2n + 1 \). This takes just a few steps to apply, making it extremely useful for larger sequences.
Utilizing this formula effectively saves time, especially when sequences grow long. It's like a shortcut that eliminates the need to repeatedly add the common difference one step at a time. By learning and practicing with explicit formulas, you'll be better equipped to handle complex arithmetic sequences with ease.
- \( a_n = a_1 + (n-1) \, d \)
The magic of the explicit formula lies in its efficiency. Once you have the first term and the common difference, you can jump directly to any term. For example, looking at the arithmetic sequence \( \{3, 5, 7, \ldots\} \), the explicit formula simplifies to \( a_n = 2n + 1 \). This takes just a few steps to apply, making it extremely useful for larger sequences.
Utilizing this formula effectively saves time, especially when sequences grow long. It's like a shortcut that eliminates the need to repeatedly add the common difference one step at a time. By learning and practicing with explicit formulas, you'll be better equipped to handle complex arithmetic sequences with ease.
Common Difference
The common difference in an arithmetic sequence is a critical component that defines the progression of the sequence. It indicates the constant amount that you add to each term to arrive at the next term.
To find the common difference, simply subtract the first term from the second term in the sequence. For example, with the sequence \( \{3, 5, 7, \ldots\} \), the second term is 5 and the first term is 3, resulting in a common difference of \( 5 - 3 = 2 \). This means that each term increases by 2.
A consistent common difference is what makes a sequence "arithmetic." It's a simple yet fundamental property. If the common difference were to change at any point, then we would no longer categorize the sequence as arithmetic.
Remember, the common difference can be positive, negative, or even zero. Positive common differences result in an increasing sequence, while negative differences make it decreasing. A common difference of zero indicates that all terms are the same. Grasping this concept helps you understand how arithmetic sequences are structured and how they grow or shrink.
To find the common difference, simply subtract the first term from the second term in the sequence. For example, with the sequence \( \{3, 5, 7, \ldots\} \), the second term is 5 and the first term is 3, resulting in a common difference of \( 5 - 3 = 2 \). This means that each term increases by 2.
A consistent common difference is what makes a sequence "arithmetic." It's a simple yet fundamental property. If the common difference were to change at any point, then we would no longer categorize the sequence as arithmetic.
Remember, the common difference can be positive, negative, or even zero. Positive common differences result in an increasing sequence, while negative differences make it decreasing. A common difference of zero indicates that all terms are the same. Grasping this concept helps you understand how arithmetic sequences are structured and how they grow or shrink.
General Formula
The general formula for an arithmetic sequence not only identifies each term precisely, but it also embodies the essence of the sequence's structure. The formula is expressed as:
In our example sequence \( \{3, 5, 7, \ldots\} \), applying the general formula involves substituting \( a_1 = 3 \) and \( d = 2 \). Performing these substitutions gives the formula \( a_n = 3 + (n-1) \cdot 2 \), which simplifies to \( a_n = 2n + 1 \).
Understanding the general formula is crucial as it provides a systematic method for determining any term in the sequence. This formula works beautifully because it accounts for the linear growth or decline dictated by the common difference. By mastering the general formula, you'll have the tools needed to expertly navigate arithmetic sequences and generate terms with confidence.
- \( a_n = a_1 + (n-1) \cdot d \)
In our example sequence \( \{3, 5, 7, \ldots\} \), applying the general formula involves substituting \( a_1 = 3 \) and \( d = 2 \). Performing these substitutions gives the formula \( a_n = 3 + (n-1) \cdot 2 \), which simplifies to \( a_n = 2n + 1 \).
Understanding the general formula is crucial as it provides a systematic method for determining any term in the sequence. This formula works beautifully because it accounts for the linear growth or decline dictated by the common difference. By mastering the general formula, you'll have the tools needed to expertly navigate arithmetic sequences and generate terms with confidence.
Other exercises in this chapter
Problem 43
For the following exercises, find the sum of the infinite geometric series. \(-1-\frac{1}{4}-\frac{1}{16}-\frac{1}{64} \ldots\)
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For the following exercises, a coin is tossed, and a card is pulled from a standard deck. Find the probability of the following: A tail on the coin or red ace
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