What Is The 50th Term Of The Arithmetic Sequence?${ 32, 27, 22, 17, 12, \ldots }$A. 7 B. 277 C. -213 D. -208
An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. In this problem, we are given an arithmetic sequence with the first five terms: 32, 27, 22, 17, 12, and so on. We are asked to find the 50th term of this sequence.
Understanding the Arithmetic Sequence
To find the 50th term of the sequence, we need to understand the pattern of the sequence. The given sequence is: 32, 27, 22, 17, 12, and so on. We can see that each term is decreasing by 5. This means that the common difference (d) between any two consecutive terms is -5.
Finding the 50th Term
To find the 50th term of the sequence, we can use the formula for the nth term of an arithmetic sequence:
an = a1 + (n - 1)d
where an is the nth term, a1 is the first term, n is the term number, and d is the common difference.
In this case, a1 = 32, n = 50, and d = -5. Plugging these values into the formula, we get:
a50 = 32 + (50 - 1)(-5) a50 = 32 + 49(-5) a50 = 32 - 245 a50 = -213
Therefore, the 50th term of the sequence is -213.
Conclusion
In this problem, we were asked to find the 50th term of an arithmetic sequence. We used the formula for the nth term of an arithmetic sequence to find the answer. The common difference between any two consecutive terms was -5, and the first term was 32. By plugging these values into the formula, we found that the 50th term of the sequence is -213.
Arithmetic Sequence Formula
The formula for the nth term of an arithmetic sequence is:
an = a1 + (n - 1)d
where an is the nth term, a1 is the first term, n is the term number, and d is the common difference.
Example Problems
Here are a few example problems to help you practice finding the nth term of an arithmetic sequence:
- Find the 20th term of the arithmetic sequence: 5, 10, 15, 20, 25, and so on.
- Find the 30th term of the arithmetic sequence: 2, 7, 12, 17, 22, and so on.
- Find the 40th term of the arithmetic sequence: 8, 3, -2, -7, -12, and so on.
Step-by-Step Solution
Here is a step-by-step solution to the problem:
- Identify the first term (a1) and the common difference (d) of the sequence.
- Plug the values of a1, n, and d into the formula for the nth term of an arithmetic sequence.
- Simplify the expression to find the value of the nth term.
Tips and Tricks
Here are a few tips and tricks to help you solve problems like this:
- Make sure to identify the first term and the common difference of the sequence.
- Use the formula for the nth term of an arithmetic sequence to find the answer.
- Simplify the expression to find the value of the nth term.
Common Mistakes
Here are a few common mistakes to avoid when solving problems like this:
- Make sure to identify the first term and the common difference of the sequence.
- Use the correct formula for the nth term of an arithmetic sequence.
- Simplify the expression to find the value of the nth term.
Conclusion
In this article, we will answer some frequently asked questions about arithmetic sequences. We will cover topics such as finding the nth term, common differences, and more.
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. For example, the sequence 2, 5, 8, 11, 14, and so on is an arithmetic sequence because the difference between any two consecutive terms is 3.
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:
an = a1 + (n - 1)d
where an is the nth term, a1 is the first term, n is the term number, and d is the common difference.
Q: What is the common difference in an arithmetic sequence?
A: The common difference in an arithmetic sequence is the difference between any two consecutive terms. For example, in the sequence 2, 5, 8, 11, 14, and so on, the common difference is 3.
Q: How do I find the common difference in an arithmetic sequence?
A: To find the common difference in an arithmetic sequence, you can subtract any term from the previous term. For example, in the sequence 2, 5, 8, 11, 14, and so on, you can subtract 2 from 5 to get 3, which is the common difference.
Q: What is the formula for the sum of an arithmetic sequence?
A: The formula for the sum of an arithmetic sequence is:
S = (n/2)(a1 + an)
where S is the sum, n is the number of terms, a1 is the first term, and an is the nth term.
Q: How do I find the sum of an arithmetic sequence?
A: To find the sum of an arithmetic sequence, you can use the formula:
S = (n/2)(a1 + an)
where S is the sum, n is the number of terms, a1 is the first term, and an is the nth term.
Q: What is the formula for the average of an arithmetic sequence?
A: The formula for the average of an arithmetic sequence is:
A = (a1 + an)/2
where A is the average, a1 is the first term, and an is the nth term.
Q: How do I find the average of an arithmetic sequence?
A: To find the average of an arithmetic sequence, you can use the formula:
A = (a1 + an)/2
where A is the average, a1 is the first term, and an is the nth term.
Q: What is the formula for the nth term of a geometric sequence?
A: The formula for the nth term of a geometric sequence is:
an = ar^(n-1)
where an is the nth term, a is the first term, r is the common ratio, and n is the term number.
Q: How do I find the nth term of a geometric sequence?
A: To find the nth term of a geometric sequence, you can use the formula:
an = ar^(n-1)
where an is the nth term, a is the first term, r is the common ratio, and n is the term number.
Q: What is the formula for the sum of a geometric sequence?
A: The formula for the sum of a geometric sequence is:
S = a(1 - r^n)/(1 - r)
where S is the sum, a is the first term, r is the common ratio, and n is the number of terms.
Q: How do I find the sum of a geometric sequence?
A: To find the sum of a geometric sequence, you can use the formula:
S = a(1 - r^n)/(1 - r)
where S is the sum, a is the first term, r is the common ratio, and n is the number of terms.
Conclusion
In this article, we answered some frequently asked questions about arithmetic sequences. We covered topics such as finding the nth term, common differences, and more. We also provided formulas and examples to help you practice finding the nth term, sum, and average of an arithmetic sequence.