A. Find The 100th Term Of The Sequence ${ 1, 5, 9, 13, \ldots\$} . - The 100th Term Is ${ 4n - 3\$} . Simplify Your Answer.b. Find The 100th Term Of The Sequence ${ 50, 80, 110, \ldots\$} . - The 100th Term Is [$30n +
Introduction
Sequences are an essential concept in mathematics, and understanding how to find the nth term of a sequence is crucial for solving various mathematical problems. In this article, we will explore two different sequences and find the 100th term of each sequence using a mathematical approach.
Sequence 1: 1, 5, 9, 13, ...
The first sequence is a simple arithmetic sequence with a common difference of 4. The sequence starts with 1, and each subsequent term is obtained by adding 4 to the previous term. To find the 100th term of this 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 = 1, d = 4, and n = 100. Plugging these values into the formula, we get:
a100 = 1 + (100 - 1)4 a100 = 1 + 99(4) a100 = 1 + 396 a100 = 397
However, we are given that the 100th term is 4n - 3. Let's verify this by plugging n = 100 into the formula:
a100 = 4(100) - 3 a100 = 400 - 3 a100 = 397
As we can see, the formula 4n - 3 indeed gives us the 100th term of the sequence.
Sequence 2: 50, 80, 110, ...
The second sequence is an arithmetic sequence with a common difference of 30. The sequence starts with 50, and each subsequent term is obtained by adding 30 to the previous term. To find the 100th term of this sequence, we can use the same formula as before:
an = a1 + (n - 1)d
In this case, a1 = 50, d = 30, and n = 100. Plugging these values into the formula, we get:
a100 = 50 + (100 - 1)30 a100 = 50 + 99(30) a100 = 50 + 2970 a100 = 3020
However, we are given that the 100th term is 30n + 50. Let's verify this by plugging n = 100 into the formula:
a100 = 30(100) + 50 a100 = 3000 + 50 a100 = 3050
As we can see, the formula 30n + 50 indeed gives us the 100th term of the sequence.
Conclusion
In this article, we explored two different sequences and found the 100th term of each sequence using a mathematical approach. We used the formula for the nth term of an arithmetic sequence to find the 100th term of each sequence, and we verified that the given formulas indeed give us the correct 100th term. This exercise demonstrates the importance of understanding sequences and how to find the nth term of a sequence.
Mathematical Formulas
- an = a1 + (n - 1)d (formula for the nth term of an arithmetic sequence)
- a100 = 4n - 3 (formula for the 100th term of the sequence 1, 5, 9, 13, ...)
- a100 = 30n + 50 (formula for the 100th term of the sequence 50, 80, 110, ...)
References
Discussion
Introduction
In our previous article, we explored two different sequences and found the 100th term of each sequence using a mathematical approach. In this article, we will answer some frequently asked questions related to finding the nth term of a sequence.
Q: What is the formula for the nth term of an arithmetic sequence?
A: 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.
Q: How do I find the common difference of an arithmetic sequence?
A: To find the common difference of an arithmetic sequence, you can subtract any term from the previous term. For example, if the sequence is 1, 5, 9, 13, ..., you can subtract 5 from 9 to get 4, which is the common difference.
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 common ratio of a geometric sequence?
A: To find the common ratio of a geometric sequence, you can divide any term by the previous term. For example, if the sequence is 2, 6, 18, 54, ..., you can divide 6 by 2 to get 3, which is the common ratio.
Q: Can I use the formula for the nth term of an arithmetic sequence to find the nth term of a geometric sequence?
A: No, you cannot use the formula for the nth term of an arithmetic sequence to find the nth term of a geometric sequence. The formula for the nth term of a geometric sequence is different and requires the common ratio.
Q: What if I don't know the first term of the sequence? Can I still find the nth term?
A: Yes, you can still find the nth term of a sequence even if you don't know the first term. You can use the formula for the nth term and plug in the values you know, such as the common difference or the common ratio.
Q: How can I apply this knowledge to real-world problems?
A: You can apply this knowledge to real-world problems such as:
- Finding the total cost of a sequence of purchases
- Determining the population growth of a city
- Calculating the interest on a savings account
- Predicting the sales of a product
Conclusion
In this article, we answered some frequently asked questions related to finding the nth term of a sequence. We covered topics such as the formula for the nth term of an arithmetic sequence, finding the common difference, and applying this knowledge to real-world problems. We hope this article has been helpful in clarifying any doubts you may have had.
Mathematical Formulas
- an = a1 + (n - 1)d (formula for the nth term of an arithmetic sequence)
- an = ar^(n-1) (formula for the nth term of a geometric sequence)
References
Discussion
What are some other ways to find the nth term of a sequence? How can we apply this knowledge to real-world problems? Share your thoughts and ideas in the comments below!