Find The Sum Of The First 8 Terms Of The Following Arithmetic Sequence, Given:$\[ A_5 = 10, \quad A_7 = 16 \\]The Sum Of The First 8 Terms Is \[$\square\$\]

Introduction

In this article, we will explore the concept of arithmetic sequences and how to find the sum of the first 8 terms of a given sequence. An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is called the common difference.

Understanding the Given Sequence

We are given the following information about the arithmetic sequence:

• $a_5 = 10$
• $a_7 = 16$

Our goal is to find the sum of the first 8 terms of this sequence.

Finding the Common Difference

To find the sum of the first 8 terms, we need to know the common difference between the terms. We can use the given information to find the common difference.

Let's denote the common difference as $d$. We know that the difference between any two consecutive terms is constant, so we can write:

$a_7 - a_5 = d$

Substituting the given values, we get:

$16 - 10 = d$

$d = 6$

So, the common difference is 6.

Finding the First Term

Now that we know the common difference, we can find the first term of the sequence. Let's denote the first term as $a_1$. We can use the formula:

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

where $n$ is the term number.

We know that $a_5 = 10$, so we can substitute $n=5$ and $d=6$ into the formula:

$10 = a_1 + (5-1)6$

Simplifying the equation, we get:

$10 = a_1 + 24$

$a_1 = -14$

So, the first term of the sequence is -14.

Finding the Sum of the First 8 Terms

Now that we know the first term and the common difference, we can find the sum of the first 8 terms. The formula for the sum of the first $n$ terms of an arithmetic sequence is:

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

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

We know that $n=8$, $a_1=-14$, and $a_8$ is the 8th term. We can find $a_8$ using the formula:

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

Substituting $n=8$, $a_1=-14$, and $d=6$, we get:

$a_8 = -14 + (8-1)6$

$a_8 = -14 + 42$

$a_8 = 28$

Now that we know $a_8$, we can find the sum of the first 8 terms:

$S_8 = \frac{8}{2}(-14 + 28)$

$S_8 = 4(14)$

$S_8 = 56$

So, the sum of the first 8 terms of the sequence is 56.

Conclusion

In this article, we found the sum of the first 8 terms of an arithmetic sequence given the 5th and 7th terms. We first found the common difference using the given terms, then found the first term using the common difference and the 5th term. Finally, we found the sum of the first 8 terms using the formula for the sum of an arithmetic sequence. The sum of the first 8 terms is 56.

Arithmetic Sequence Formula

The formula for the sum of the first $n$ terms of an arithmetic sequence is:

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

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

Arithmetic Sequence Example

Find the sum of the first 8 terms of the arithmetic sequence given:

$a_5 = 10$ $a_7 = 16$

Solution

1. Find the common difference using the given terms:

$d = a_7 - a_5$ $d = 16 - 10$ $d = 6$

2. Find the first term using the common difference and the 5th term:

$a_n = a_1 + (n-1)d$ $10 = a_1 + (5-1)6$ $10 = a_1 + 24$ $a_1 = -14$

3. Find the 8th term using the formula:

$a_n = a_1 + (n-1)d$ $a_8 = -14 + (8-1)6$ $a_8 = -14 + 42$ $a_8 = 28$

4. Find the sum of the first 8 terms using the formula:

$S_n = \frac{n}{2}(a_1 + a_n)$ $S_8 = \frac{8}{2}(-14 + 28)$ $S_8 = 4(14)$ $S_8 = 56$

Q: What is an arithmetic sequence?

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

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

A: To find the common difference, you can use the formula:

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

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

Alternatively, you can use the given terms to find the common difference. For example, if you are given the 5th and 7th terms, you can use the formula:

$d = a_7 - a_5$

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

A: To find the first term, you can use the formula:

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

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

Alternatively, you can use the given terms to find the first term. For example, if you are given the 5th term and the common difference, you can use the formula:

$a_1 = a_5 - (5-1)d$

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

A: To find the sum of the first $n$ terms, you can use the formula:

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

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

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

A: The formula for the $n$th term of an arithmetic sequence is:

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

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

Q: How do I find the nth term of an arithmetic sequence if I know the first term and the common difference?

A: To find the $n$th term, you can use the formula:

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

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

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

$S = \frac{a_1 + a_n}{2}$

where $S$ is the sum of the infinite sequence, $a_1$ is the first term, and $a_n$ is the $n$th term.

However, this formula only works if the sequence is convergent, meaning that the terms of the sequence get arbitrarily close to a finite limit as $n$ approaches infinity.

Q: What is the formula for the sum of the first n terms of an arithmetic sequence in terms of the first term and the common difference?

A: The formula for the sum of the first $n$ terms of an arithmetic sequence in terms of the first term and the common difference is:

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

where $S_n$ is the sum of the first $n$ terms, $a_1$ is the first term, and $d$ is the common difference.

Q: How do I find the sum of the first n terms of an arithmetic sequence if I know the first term and the common difference?

A: To find the sum of the first $n$ terms, you can use the formula:

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

Q: What is the formula for the nth term of an arithmetic sequence in terms of the first term and the common difference?

A: The formula for the $n$th term of an arithmetic sequence in terms of the first term and the common difference is:

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

Q: How do I find the nth term of an arithmetic sequence if I know the first term and the common difference?

A: To find the $n$th term, you can use the formula:

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

Q: What is the relationship between the sum of the first n terms and the sum of the first (n-1) terms of an arithmetic sequence?

A: The sum of the first $n$ terms of an arithmetic sequence is equal to the sum of the first $(n-1)$ terms plus the $n$th term:

$S_n = S_{n-1} + a_n$

Q: How do I find the sum of the first n terms of an arithmetic sequence if I know the sum of the first (n-1) terms and the nth term?

A: To find the sum of the first $n$ terms, you can use the formula:

$S_n = S_{n-1} + a_n$

Q: What is the formula for the sum of the first n terms of an arithmetic sequence in terms of the first term, the common difference, and the number of terms?

A: The formula for the sum of the first $n$ terms of an arithmetic sequence in terms of the first term, the common difference, and the number of terms is:

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

Q: How do I find the sum of the first n terms of an arithmetic sequence if I know the first term, the common difference, and the number of terms?

A: To find the sum of the first $n$ terms, you can use the formula:

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