Select The Correct Answer.What Is The Sum Of The First 17 Terms Of This Arithmetic Series? Use $S_n=n\left(\frac{a_1+a_n}{2}\right$\] Series: \[$-28 + (-13) + 2 + 17 + \cdots\$\]A. 1,692 B. 1,564 C. 227 D. 212

Understanding Arithmetic Series

An arithmetic series is a sequence of numbers in which the difference between any two consecutive terms is constant. The general formula for the nth term of an arithmetic series is given by:

$$a_n = a_1 + (n-1)d$$

where $a_n$ is the nth term, $a_1$ is the first term, $n$ is the term number, and $d$ is the common difference.

Given Series

The given series is:

$$-28 + (-13) + 2 + 17 + \cdots$$

We need to find the sum of the first 17 terms of this series.

Finding the Common Difference

To find the common difference, we can subtract any two consecutive terms. Let's subtract the second term from the first term:

$$(-13) - (-28) = 15$$

So, the common difference is 15.

Finding the Sum of the First 17 Terms

Now that we have the common difference, we can use the formula for the sum of an arithmetic series:

$$S_n = n\left(\frac{a_1+a_n}{2}\right)$$

where $S_n$ is the sum of the first n terms, $n$ is the number of terms, $a_1$ is the first term, and $a_n$ is the nth term.

To find the sum of the first 17 terms, we need to find the 17th term. We can use the formula for the nth term:

$$a_n = a_1 + (n-1)d$$

where $a_n$ is the nth term, $a_1$ is the first term, $n$ is the term number, and $d$ is the common difference.

Plugging in the values, we get:

$$a_{17} = -28 + (17-1)15$$

$$a_{17} = -28 + 16(15)$$

$$a_{17} = -28 + 240$$

$$a_{17} = 212$$

Now that we have the 17th term, we can find the sum of the first 17 terms:

$$S_{17} = 17\left(\frac{-28+212}{2}\right)$$

$$S_{17} = 17\left(\frac{184}{2}\right)$$

$$S_{17} = 17(92)$$

$$S_{17} = 1564$$

Conclusion

The sum of the first 17 terms of the given arithmetic series is 1564.

Answer

The correct answer is:

• B. 1,564

Discussion

This problem requires the application of the formula for the sum of an arithmetic series. The common difference is found by subtracting any two consecutive terms, and the 17th term is found using the formula for the nth term. The sum of the first 17 terms is then found by plugging in the values into the formula for the sum of an arithmetic series.

Related Topics

• Arithmetic series
• Geometric series
• Sum of an arithmetic series
• Formula for the nth term of an arithmetic series

Practice Problems

1. Find the sum of the first 10 terms of the arithmetic series: 2, 5, 8, 11, ...
2. Find the sum of the first 15 terms of the arithmetic series: -3, 0, 3, 6, ...
3. Find the sum of the first 20 terms of the arithmetic series: 10, 15, 20, 25, ...
   Arithmetic Series: Q&A ==========================

Q: What is an arithmetic series?

A: An arithmetic series is a sequence of numbers in which the difference between any two consecutive terms is constant.

Q: What is the formula for the nth term of an arithmetic series?

A: The formula for the nth term of an arithmetic series is given by:

$$a_n = a_1 + (n-1)d$$

where $a_n$ is the nth term, $a_1$ is the first term, $n$ is the term number, and $d$ is the common difference.

Q: How do I find the common difference of an arithmetic series?

A: To find the common difference, you can subtract any two consecutive terms. For example, if the series is 2, 5, 8, 11, ..., you can subtract the second term from the first term:

$$5 - 2 = 3$$

So, the common difference is 3.

Q: What is the formula for the sum of an arithmetic series?

A: The formula for the sum of an arithmetic series is given by:

$$S_n = n\left(\frac{a_1+a_n}{2}\right)$$

where $S_n$ is the sum of the first n terms, $n$ is the number of terms, $a_1$ is the first term, and $a_n$ is the nth term.

Q: How do I find the sum of the first n terms of an arithmetic series?

A: To find the sum of the first n terms, you need to find the nth term using the formula for the nth term:

$$a_n = a_1 + (n-1)d$$

Then, you can plug in the values into the formula for the sum of an arithmetic series:

$$S_n = n\left(\frac{a_1+a_n}{2}\right)$$

Q: What is the difference between an arithmetic series and a geometric series?

A: An arithmetic series is a sequence of numbers in which the difference between any two consecutive terms is constant. A geometric series is a sequence of numbers in which the ratio between any two consecutive terms is constant.

Q: How do I find the sum of an infinite arithmetic series?

A: To find the sum of an infinite arithmetic series, you need to find the common difference and the first term. Then, you can use the formula for the sum of an infinite arithmetic series:

$$S = \frac{a_1}{1-r}$$

where $S$ is the sum, $a_1$ is the first term, and $r$ is the common ratio.

Q: What is the formula for the sum of an infinite geometric series?

A: The formula for the sum of an infinite geometric series is given by:

$$S = \frac{a_1}{1-r}$$

where $S$ is the sum, $a_1$ is the first term, and $r$ is the common ratio.

Q: How do I determine if an arithmetic series is convergent or divergent?

A: An arithmetic series is convergent if the common difference is negative, and divergent if the common difference is positive.

Q: How do I determine if a geometric series is convergent or divergent?

A: A geometric series is convergent if the common ratio is between -1 and 1, and divergent if the common ratio is outside this range.

Q: What are some real-world applications of arithmetic series?

A: Arithmetic series have many real-world applications, including:

• Finance: calculating interest rates and investment returns
• Music: calculating the frequency of notes in a musical scale
• Physics: calculating the motion of objects under constant acceleration
• Engineering: calculating the stress and strain on materials under load

Q: What are some real-world applications of geometric series?

A: Geometric series have many real-world applications, including:

• Finance: calculating compound interest and investment returns
• Music: calculating the frequency of notes in a musical scale
• Physics: calculating the motion of objects under constant acceleration
• Engineering: calculating the stress and strain on materials under load