Problem 90
Question
Identifying a Sequence Determine whether the sequence associated with the series is arithmetic or geometric. Find the common difference or ratio and find the sum of the first 15 terms. $$17+14+11+8+\cdot \cdot \cdot \cdot$$
Step-by-Step Solution
Verified Answer
The sequence is an arithmetic sequence with a common difference of -3. The sum of the first 15 terms of the sequence is -75
1Step 1: Identify the type of sequence
Examine the sequence and notice that each term is decreasing by the same amount, 3. Therefore, this is an arithmetic sequence.
2Step 2: Determine the common difference
In an arithmetic sequence, the common difference \(d\) is the amount by which each term decreases from one term to the next. Here, each term decreases by 3, so \(d = -3\). This is also confirmed by subtracting any term from the one preceding it, which also gives a difference of -3.
3Step 3: Use the formula for the sum of an arithmetic series
The sum \(S_n\) of the first \(n\) terms of an arithmetic series can be found using the formula \(S_n = n/2 * (a_1 + a_n)\), where \(a_1\) is the first term and \(a_n\) is the nth term. In this problem, we need to find the sum of the first 15 terms, so \(n = 15\), \(a_1 = 17\), and \(a_{15} = 17 + (15-1)*(-3) = -27\).
4Step 4: Calculate the sum of the first 15 terms
Using the formula and the values found: \(S_{15} = 15/2 * (17 - 27) = -75\).
Key Concepts
Common DifferenceArithmetic SeriesSum of Arithmetic Series
Common Difference
In an arithmetic sequence, the 'common difference' is a key concept that serves as the foundation for understanding these types of sequences. The common difference (\( d \)) is the constant amount added or subtracted from one term to get to the next. For instance, consider the sequence \( 5, 8, 11, 14, \ldots \), where each term is three more than the previous one. The common difference here is \( +3 \).
In the exercise provided, we are given an arithmetic series: \( 17, 14, 11, \ldots \). To find the common difference, one method is to subtract the second term from the first term, resulting in \( 14 - 17 = -3 \). This means that the sequence is decreasing by the same amount, which is \( d = -3 \). Identifying the common difference is critical, as it allows us to predict future terms in the sequence and is used to calculate the sum of an arithmetic series.
In the exercise provided, we are given an arithmetic series: \( 17, 14, 11, \ldots \). To find the common difference, one method is to subtract the second term from the first term, resulting in \( 14 - 17 = -3 \). This means that the sequence is decreasing by the same amount, which is \( d = -3 \). Identifying the common difference is critical, as it allows us to predict future terms in the sequence and is used to calculate the sum of an arithmetic series.
Arithmetic Series
An arithmetic series is essentially the sum of the terms of an arithmetic sequence. When you look at a list of numbers like \( 2, 4, 6, 8, \ldots \), and if these numbers are being added together, such as \( 2 + 4 + 6 + 8 + \ldots \), you're dealing with an arithmetic series. The sequence has a common difference, and in the case of an arithmetic series, this common difference is used to determine the sum of a specified number of terms.
In our exercise, the series \( 17+14+11+8+ \ldots \cdot \ldots \) represents the sum of arithmetic sequence terms. To solve problems involving arithmetic series, it's important to first identify that the series is arithmetic by confirming a constant common difference between terms, like the \( -3 \) we calculated previously.
In our exercise, the series \( 17+14+11+8+ \ldots \cdot \ldots \) represents the sum of arithmetic sequence terms. To solve problems involving arithmetic series, it's important to first identify that the series is arithmetic by confirming a constant common difference between terms, like the \( -3 \) we calculated previously.
Sum of Arithmetic Series
Once you've identified an arithmetic series and its common difference, you may want to find the sum of a certain number of terms in that series, known as the 'sum of arithmetic series'. There's a handy formula to calculate this: \[ S_n = \frac{n}{2} * (a_1 + a_n) \], where \( S_n \) is the sum of the first \( n \) terms, \( a_1 \) is the first term, and \( a_n \) is the nth term.
In our exercise, we need the sum of the first 15 terms. Using the formula, we find the 15th term by using the first term \( 17 \) and the common difference \( -3 \) to get \( a_{15} = 17 + (15-1)*(-3) = -27 \). Then, we apply the formula \( S_{15} = \frac{15}{2} * (17 - 27) \) to get the sum \( S_{15} = -75 \). This step-by-step manipulation of the formula demonstrates how algebra is used to sum up a series systematically rather than adding up all terms individually, which can be overbearing and time-consuming, especially when dealing with a large number of terms.
In our exercise, we need the sum of the first 15 terms. Using the formula, we find the 15th term by using the first term \( 17 \) and the common difference \( -3 \) to get \( a_{15} = 17 + (15-1)*(-3) = -27 \). Then, we apply the formula \( S_{15} = \frac{15}{2} * (17 - 27) \) to get the sum \( S_{15} = -75 \). This step-by-step manipulation of the formula demonstrates how algebra is used to sum up a series systematically rather than adding up all terms individually, which can be overbearing and time-consuming, especially when dealing with a large number of terms.
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