Westonci.ca connects you with experts who provide insightful answers to your questions. Join us today and start learning! Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals.
Sagot :
To find the sum of the first 15 terms of the arithmetic series \(3 + 6 + 9 + \cdots\), we can use the formula for the sum of the first \(n\) terms of an arithmetic series:
[tex]\[ S_n = \frac{n}{2} \left(2a + (n - 1)d\right) \][/tex]
where:
- \( S_n \) is the sum of the first \( n \) terms,
- \( n \) is the number of terms,
- \( a \) is the first term of the series,
- \( d \) is the common difference between the terms.
Given the series \( 3 + 6 + 9 + \cdots \):
- The first term \( a \) is 3.
- The common difference \( d \) is 3.
- The number of terms \( n \) is 15.
Plugging these values into the formula, we get:
[tex]\[ S_{15} = \frac{15}{2} \left(2 \times 3 + (15 - 1) \times 3\right) \][/tex]
Let's break it down step by step:
1. Calculate \( 2a \):
[tex]\[ 2 \times 3 = 6 \][/tex]
2. Calculate \( (n - 1) \):
[tex]\[ 15 - 1 = 14 \][/tex]
3. Calculate \( (n - 1) \times d \):
[tex]\[ 14 \times 3 = 42 \][/tex]
4. Add \( 2a \) and \( (n - 1) \times d \):
[tex]\[ 6 + 42 = 48 \][/tex]
5. Multiply by \( \frac{n}{2} \):
[tex]\[ \frac{15}{2} \times 48 = \frac{15 \times 48}{2} = \frac{720}{2} = 360 \][/tex]
Therefore, the sum of the first 15 terms of the series is:
[tex]\[ S_{15} = 360 \][/tex]
[tex]\[ S_n = \frac{n}{2} \left(2a + (n - 1)d\right) \][/tex]
where:
- \( S_n \) is the sum of the first \( n \) terms,
- \( n \) is the number of terms,
- \( a \) is the first term of the series,
- \( d \) is the common difference between the terms.
Given the series \( 3 + 6 + 9 + \cdots \):
- The first term \( a \) is 3.
- The common difference \( d \) is 3.
- The number of terms \( n \) is 15.
Plugging these values into the formula, we get:
[tex]\[ S_{15} = \frac{15}{2} \left(2 \times 3 + (15 - 1) \times 3\right) \][/tex]
Let's break it down step by step:
1. Calculate \( 2a \):
[tex]\[ 2 \times 3 = 6 \][/tex]
2. Calculate \( (n - 1) \):
[tex]\[ 15 - 1 = 14 \][/tex]
3. Calculate \( (n - 1) \times d \):
[tex]\[ 14 \times 3 = 42 \][/tex]
4. Add \( 2a \) and \( (n - 1) \times d \):
[tex]\[ 6 + 42 = 48 \][/tex]
5. Multiply by \( \frac{n}{2} \):
[tex]\[ \frac{15}{2} \times 48 = \frac{15 \times 48}{2} = \frac{720}{2} = 360 \][/tex]
Therefore, the sum of the first 15 terms of the series is:
[tex]\[ S_{15} = 360 \][/tex]
Thank you for choosing our service. We're dedicated to providing the best answers for all your questions. Visit us again. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Thank you for choosing Westonci.ca as your information source. We look forward to your next visit.