The Explicit Rule For An Arithmetic Sequence Is A N = 20 3 + 1 3 ( N − 1 A_n = \frac{20}{3} + \frac{1}{3}(n-1 A N ​ = 3 20 ​ + 3 1 ​ ( N − 1 ]. What Is The Value Of The 89th Term?A. 36 B. 257 3 \frac{257}{3} 3 257 ​ C. 89 3 \frac{89}{3} 3 89 ​ D. 248

Introduction

An arithmetic sequence is a sequence of numbers in which the difference between any two successive members is constant. The explicit rule for an arithmetic sequence is given by 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 term number, and $d$ is the common difference. In this article, we will use the explicit rule to find the value of the 89th term of an arithmetic sequence with the given formula $a_n = \frac{20}{3} + \frac{1}{3}(n-1)$.

Understanding the Explicit Rule

The explicit rule for an arithmetic sequence is given by the formula $a_n = a_1 + (n-1)d$. This formula states that the nth term of an arithmetic sequence is equal to the first term plus the product of the common difference and the term number minus one. In other words, the nth term is equal to the first term plus the sum of the common difference multiplied by the number of terms minus one.

Applying the Explicit Rule to the Given Sequence

The explicit rule for the given arithmetic sequence is $a_n = \frac{20}{3} + \frac{1}{3}(n-1)$. To find the value of the 89th term, we need to substitute $n=89$ into the formula.

Calculating the 89th Term

To calculate the 89th term, we need to substitute $n=89$ into the formula $a_n = \frac{20}{3} + \frac{1}{3}(n-1)$. This gives us:

$a_{89} = \frac{20}{3} + \frac{1}{3}(89-1)$

$a_{89} = \frac{20}{3} + \frac{1}{3}(88)$

$a_{89} = \frac{20}{3} + \frac{88}{3}$

$a_{89} = \frac{108}{3}$

$a_{89} = 36$

Conclusion

In this article, we used the explicit rule for an arithmetic sequence to find the value of the 89th term. We substituted $n=89$ into the formula $a_n = \frac{20}{3} + \frac{1}{3}(n-1)$ and calculated the value of the 89th term. The result is $a_{89} = 36$. This means that the 89th term of the arithmetic sequence is 36.

Step-by-Step Solution

1. Write down the explicit rule for the arithmetic sequence: $a_n = \frac{20}{3} + \frac{1}{3}(n-1)$.
2. Substitute $n=89$ into the formula: $a_{89} = \frac{20}{3} + \frac{1}{3}(89-1)$.
3. Simplify the expression: $a_{89} = \frac{20}{3} + \frac{1}{3}(88)$.
4. Combine the fractions: $a_{89} = \frac{20}{3} + \frac{88}{3}$.
5. Add the fractions: $a_{89} = \frac{108}{3}$.
6. Simplify the fraction: $a_{89} = 36$.

Final Answer

Introduction

In our previous article, we used the explicit rule for an arithmetic sequence to find the value of the 89th term. In this article, we will answer some frequently asked questions about the explicit rule for an arithmetic sequence.

Q: What is the explicit rule for an arithmetic sequence?

A: The explicit rule for an arithmetic sequence is given by 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 term number, and $d$ is the common difference.

Q: How do I use the explicit rule to find the nth term of an arithmetic sequence?

A: To use the explicit rule to find the nth term of an arithmetic sequence, you need to substitute the values of $a_1$, $n$, and $d$ into the formula $a_n = a_1 + (n-1)d$. Then, simplify the expression to find the value of the nth term.

Q: What is the difference between the explicit rule and the recursive rule for an arithmetic sequence?

A: The explicit rule for an arithmetic sequence is given by the formula $a_n = a_1 + (n-1)d$, while the recursive rule is given by the formula $a_n = a_{n-1} + d$. The explicit rule gives the value of the nth term directly, while the recursive rule gives the value of the nth term in terms of the previous term.

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

A: To find the common difference of an arithmetic sequence, you need to find the difference between any two successive terms. For example, if the first term is $a_1$ and the second term is $a_2$, the common difference is given by $d = a_2 - a_1$.

Q: Can I use the explicit rule to find the first term of an arithmetic sequence?

A: Yes, you can use the explicit rule to find the first term of an arithmetic sequence. If you know the value of the nth term, the common difference, and the term number, you can substitute these values into the formula $a_n = a_1 + (n-1)d$ and solve for $a_1$.

Q: What are some real-world applications of the explicit rule for an arithmetic sequence?

A: The explicit rule for an arithmetic sequence has many real-world applications, such as:

• Modeling population growth or decline
• Calculating interest rates or investment returns
• Analyzing data from experiments or surveys
• Predicting future values of a sequence

Conclusion

In this article, we answered some frequently asked questions about the explicit rule for an arithmetic sequence. We hope that this article has helped you to better understand the explicit rule and how to use it to solve problems.

Step-by-Step Solution

1. Write down the explicit rule for the arithmetic sequence: $a_n = a_1 + (n-1)d$.
2. Substitute the values of $a_1$, $n$, and $d$ into the formula.
3. Simplify the expression to find the value of the nth term.
4. Use the explicit rule to find the first term of the arithmetic sequence.
5. Use the explicit rule to find the common difference of the arithmetic sequence.

Final Answer

The final answer is $\boxed{36}$.