Finding the Number of Terms

Oct 13, 2025 by windailys.com 58 views

Introduction

In mathematics, a series is a sequence of numbers that are added together to find a total sum. The series we are dealing with in this article is an arithmetic sequence, where each term is obtained by adding a fixed constant to the previous term. In this case, the series is given as: $6.5, 8, 9.5, 11.0, \ldots, 32$ . Our goal is to determine the sum of this series.

Understanding the Series

The given series is an arithmetic sequence with a common difference of $1.5$ . This means that each term is obtained by adding $1.5$ to the previous term. To find the sum of the series, we need to find the number of terms in the series and then use the formula for the sum of an arithmetic series.

Finding the Number of Terms

To find the number of terms in the series, we can use the formula for the nth term of an arithmetic sequence:

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

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

In this case, we know that the last term is $32$ , the first term is $6.5$ , and the common difference is $1.5$ . We can plug these values into the formula to find the number of terms:

$32 = 6.5 + (n-1)1.5$

Simplifying the equation, we get:

$25.5 = (n-1)1.5$

Dividing both sides by $1.5$ , we get:

$17 = n-1$

Adding $1$ to both sides, we get:

$n = 18$

Therefore, the series has $18$ terms.

Finding the Sum of the Series

Now that we know the number of terms in the series, we can use the formula for the sum of an arithmetic series:

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

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

Plugging in the values, we get:

$S_{18} = \frac{18}{2}(6.5 + 32)$

Simplifying the equation, we get:

$S_{18} = 9(38.5)$

Multiplying the numbers, we get:

$S_{18} = 346.5$

Therefore, the sum of the series is $346.5$ .

Conclusion

In this article, we determined the sum of the series $6.5, 8, 9.5, 11.0, \ldots, 32$ . We first found the number of terms in the series by using the formula for the nth term of an arithmetic sequence. Then, we used the formula for the sum of an arithmetic series to find the sum of the series. The sum of the series is $346.5$ .

Arithmetic Series Formula

The formula for the sum of an arithmetic series is:

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

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

Arithmetic Sequence Formula

The formula for the nth term of an arithmetic sequence is:

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

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

Common Difference

The common difference is the fixed constant that is added to each term to get the next term. In this case, the common difference is $1.5$ .

Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which each term is obtained by adding a fixed constant to the previous term. In this case, the arithmetic sequence is $6.5, 8, 9.5, 11.0, \ldots, 32$ .

Arithmetic Series

An arithmetic series is a sequence of numbers in which each term is obtained by adding a fixed constant to the previous term. In this case, the arithmetic series is $6.5, 8, 9.5, 11.0, \ldots, 32$ .

Mathematical Formulas

The mathematical formulas used in this article are:

$a_n = a_1 + (n-1)d$
$S_n = \frac{n}{2}(a_1 + a_n)$

These formulas are used to find the nth term of an arithmetic sequence and the sum of an arithmetic series, respectively.

Conclusion

$In this article, we determined the sum of the series $6.5, 8, 9.5, 11.0, \ldots, 32$. We first found the number of terms in the series by using the formula for the nth term of an arithmetic sequence. Then, we used the formula for the sum of an arithmetic series to find the sum of the series. The sum of the series is $346.5$.<br/> **Determine the Sum of the Series: A Mathematical Approach**$

Q&A: Determining the Sum of the Series

Q: What is an arithmetic sequence?

A: An arithmetic sequence is a sequence of numbers in which each term is obtained by adding a fixed constant to the previous term.

Q: What is the common difference in an arithmetic sequence?

A: The common difference is the fixed constant that is added to each term to get the next term.

Q: How do I find the nth term of an arithmetic sequence?

A: To find the nth term of an arithmetic sequence, you can use the formula:

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

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

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

A: To find the sum of an arithmetic series, you can use the formula:

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

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

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

A: An arithmetic sequence is a sequence of numbers in which each term is obtained by adding a fixed constant to the previous term. An arithmetic series is a sequence of numbers in which each term is obtained by adding a fixed constant to the previous term, and the sum of the series is calculated.

Q: How do I determine the number of terms in an arithmetic sequence?

A: To determine the number of terms in an arithmetic sequence, you can use the formula:

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

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

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

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

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

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

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

A: To calculate the sum of an arithmetic series, you can use the formula:

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

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

Q: What is the significance of the common difference in an arithmetic sequence?

A: The common difference is the fixed constant that is added to each term to get the next term. It is an important concept in arithmetic sequences and series.

Q: How do I determine the common difference in an arithmetic sequence?

A: To determine the common difference in an arithmetic sequence, you can use the formula:

$d = a_n - a_{n-1}$

where $d$ is the common difference, $a_n$ is the nth term, and $a_{n-1}$ is the (n-1)th term.

Q: What is the relationship between the sum of an arithmetic series and the number of terms?

A: The sum of an arithmetic series is directly proportional to the number of terms. The more terms there are, the larger the sum will be.

Q: How do I use the formula for the sum of an arithmetic series to find the sum of a given series?

A: To use the formula for the sum of an arithmetic series to find the sum of a given series, you need to know the first term, the last term, and the number of terms. Then, you can plug these values into the formula:

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

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

Conclusion

In this article, we have answered some of the most frequently asked questions about determining the sum of an arithmetic series. We have covered topics such as arithmetic sequences, common differences, and the formula for the sum of an arithmetic series. We hope that this article has been helpful in understanding the concepts of arithmetic sequences and series.