If one term in the arithmetic sequence is a21 = -17 and the common difference is d = - 3 Find the following:

a) Write a rule that can find any term in the sequence.
•b) Find the twelfth term (a12) and
eighty-second term (a82) term.

In an arithmetic sequence, the \( n \)-th term \( a_n \) can be found using the formula:

\[ a_n = a_1 + (n - 1)d \]

where \( a_1 \) is the first term and \( d \) is the common difference.

### Part (a): Write a Rule to Find Any Term in the Sequence
Given that \( a_{21} = -17 \) and \( d = -3 \), we can use this information to find the first term \( a_1 \).

Using the formula for the \( n \)-th term:
\[ a_{21} = a_1 + (21 - 1)d \]
\[ -17 = a_1 + 20(-3) \]
\[ -17 = a_1 - 60 \]
\[ a_1 = -17 + 60 \]
\[ a_1 = 43 \]

Now we can write the rule for the \( n \)-th term:
\[ a_n = 43 + (n - 1)(-3) \]
\[ a_n = 43 - 3(n - 1) \]
\[ a_n = 43 - 3n + 3 \]
\[ a_n = 46 - 3n \]

### Part (b): Find the Twelfth Term (\( a_{12} \)) and Eighty-second Term (\( a_{82} \))
Using the rule \( a_n = 46 - 3n \):

1. **Twelfth term (\( a_{12} \))**:
\[ a_{12} = 46 - 3(12) \]
\[ a_{12} = 46 - 36 \]
\[ a_{12} = 10 \]

2. **Eighty-second term (\( a_{82} \))**:
\[ a_{82} = 46 - 3(82) \]
\[ a_{82} = 46 - 246 \]
\[ a_{82} = -200 \]

So, the rule for any term in the sequence is \( a_n = 46 - 3n \), the twelfth term is 10, and the eighty-second term is -200.

Would you like more details or have any questions about these calculations?

Here are some related questions to expand your understanding:

1. How do you derive the general formula for the \( n \)-th term of an arithmetic sequence?
2. What is the significance of the common difference in an arithmetic sequence?
3. How can you find the first term if only the \( n \)-th term and common difference are given?
4. What happens to the terms of an arithmetic sequence if the common difference is positive?
5. How do you determine the sum of the first \( n \) terms in an arithmetic sequence?
6. What is the relationship between the terms in an arithmetic sequence and linear functions?
7. Can an arithmetic sequence have a common difference of zero? What would the sequence look like?
8. How can you use the arithmetic sequence formula to solve real-world problems?

**Tip:** In arithmetic sequences, knowing any term and the common difference allows you to reconstruct the entire sequence.

Explore how to find specific terms and derive the rule for an arithmetic sequence. Learn the formula for the nth term and solve for the twelfth and eighty-second terms given specific conditions.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation