Find The 100th And The Nth Term For Each Of The Following Sequences:a. Sequence: 1 , 5 , 9 , 13 , … 1, 5, 9, 13, \ldots 1 , 5 , 9 , 13 , … B. Sequence: 50 , 80 , 110 , … 50, 80, 110, \ldots 50 , 80 , 110 , … C. Sequence: 1 , 3 , 9 , … 1, 3, 9, \ldots 1 , 3 , 9 , … D. Sequence: 9 , 9 4 , 9 7 , 9 10 , … 9, 9^4, 9^7, 9^{10}, \ldots 9 , 9 4 , 9 7 , 9 10 , … E.

In mathematics, sequences are an essential concept used to represent a list of numbers in a specific order. Sequences can be defined by a common difference, ratio, or other mathematical operations. In this article, we will explore five different sequences and find the 100th and the nth term for each of them.

Sequence a: $1, 5, 9, 13, \ldots$

The given sequence is an arithmetic sequence with a common difference of 4. To find the 100th term, we can use the formula for the nth term of an arithmetic sequence:

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

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

For this sequence, $a_1 = 1$ and $d = 4$. Plugging in $n = 100$, we get:

$$a_{100} = 1 + (100-1)4 = 1 + 399 = 400$$

Therefore, the 100th term of this sequence is 400.

To find the nth term, we can use the same formula:

$$a_n = 1 + (n-1)4$$

This formula represents the nth term of the sequence.

Sequence b: $50, 80, 110, \ldots$

The given sequence is an arithmetic sequence with a common difference of 30. To find the 100th term, we can use the formula for the nth term of an arithmetic sequence:

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

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

For this sequence, $a_1 = 50$ and $d = 30$. Plugging in $n = 100$, we get:

$$a_{100} = 50 + (100-1)30 = 50 + 2970 = 3020$$

Therefore, the 100th term of this sequence is 3020.

To find the nth term, we can use the same formula:

$$a_n = 50 + (n-1)30$$

This formula represents the nth term of the sequence.

Sequence c: $1, 3, 9, \ldots$

The given sequence is a geometric sequence with a common ratio of 3. To find the 100th term, we can use the formula for the nth term of a geometric sequence:

$$a_n = a_1 \cdot r^{n-1}$$

where $a_n$ is the nth term, $a_1$ is the first term, $r$ is the common ratio, and $n$ is the term number.

For this sequence, $a_1 = 1$ and $r = 3$. Plugging in $n = 100$, we get:

$$a_{100} = 1 \cdot 3^{100-1} = 3^{99}$$

Therefore, the 100th term of this sequence is $3^{99}$.

To find the nth term, we can use the same formula:

$$a_n = 1 \cdot 3^{n-1} = 3^{n-1}$$

This formula represents the nth term of the sequence.

Sequence d: $9, 9^4, 9^7, 9^{10}, \ldots$

The given sequence is a geometric sequence with a common ratio of 9. To find the 100th term, we can use the formula for the nth term of a geometric sequence:

$$a_n = a_1 \cdot r^{n-1}$$

where $a_n$ is the nth term, $a_1$ is the first term, $r$ is the common ratio, and $n$ is the term number.

For this sequence, $a_1 = 9$ and $r = 9$. Plugging in $n = 100$, we get:

$$a_{100} = 9 \cdot 9^{100-1} = 9^{100}$$

Therefore, the 100th term of this sequence is $9^{100}$.

To find the nth term, we can use the same formula:

$$a_n = 9 \cdot 9^{n-1} = 9^n$$

This formula represents the nth term of the sequence.

Sequence e: [Insert sequence]

Unfortunately, the problem statement does not provide a sequence for part e. However, we can still discuss the general approach to finding the 100th and the nth term of a sequence.

In general, to find the 100th term of a sequence, we can use the formula for the nth term of the sequence. If the sequence is arithmetic, we can use the formula:

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

If the sequence is geometric, we can use the formula:

$$a_n = a_1 \cdot r^{n-1}$$

where $a_n$ is the nth term, $a_1$ is the first term, $r$ is the common ratio, and $d$ is the common difference.

To find the nth term, we can use the same formula:

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

or

$$a_n = a_1 \cdot r^{n-1}$$

depending on whether the sequence is arithmetic or geometric.

In conclusion, finding the 100th and the nth term of a sequence requires a clear understanding of the sequence's properties, such as its type (arithmetic or geometric) and its common difference or ratio. By using the appropriate formula, we can easily find the 100th and the nth term of a sequence.

Key Takeaways

• To find the 100th term of an arithmetic sequence, use the formula: $a_n = a_1 + (n-1)d$
• To find the 100th term of a geometric sequence, use the formula: $a_n = a_1 \cdot r^{n-1}$
• To find the nth term of an arithmetic sequence, use the formula: $a_n = a_1 + (n-1)d$
• To find the nth term of a geometric sequence, use the formula: $a_n = a_1 \cdot r^{n-1}$

Practice Problems

1. Find the 100th term of the sequence: $2, 6, 10, 14, \ldots$
2. Find the 100th term of the sequence: $3, 9, 27, 81, \ldots$
3. Find the nth term of the sequence: $4, 8, 12, 16, \ldots$
4. Find the nth term of the sequence: $5, 25, 125, 625, \ldots$

Solutions

1. $a_{100} = 2 + (100-1)4 = 2 + 396 = 398$
2. $a_{100} = 3 \cdot 3^{100-1} = 3^{99}$
3. $a_n = 4 + (n-1)4 = 4 + 4n - 4 = 4n$
4. $a_n = 5 \cdot 5^{n-1} = 5^n$
   Q&A: Finding the 100th and the nth Term of Various Sequences ===========================================================

In our previous article, we explored five different sequences and found the 100th and the nth term for each of them. In this article, we will answer some frequently asked questions about finding the 100th and the nth term of various sequences.

Q: What is the difference between an arithmetic sequence and a geometric sequence?

A: An arithmetic sequence is a sequence of numbers in which each term after the first is obtained by adding a fixed constant to the previous term. For example, the sequence $2, 6, 10, 14, \ldots$ is an arithmetic sequence with a common difference of 4. A geometric sequence, on the other hand, is a sequence of numbers in which each term after the first is obtained by multiplying the previous term by a fixed constant. For example, the sequence $3, 9, 27, 81, \ldots$ is a geometric sequence with a common ratio of 3.

Q: How do I determine whether a sequence is arithmetic or geometric?

A: To determine whether a sequence is arithmetic or geometric, you can look at the difference between consecutive terms. If the difference is constant, the sequence is arithmetic. If the ratio between consecutive terms is constant, the sequence is geometric.

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

A: The formula for finding the nth term of an arithmetic sequence is:

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

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: What is the formula for finding the nth term of a geometric sequence?

A: The formula for finding the nth term of a geometric sequence is:

$$a_n = a_1 \cdot r^{n-1}$$

where $a_n$ is the nth term, $a_1$ is the first term, $r$ is the common ratio, and $n$ is the term number.

Q: How do I find the 100th term of a sequence?

A: To find the 100th term of a sequence, you can use the formula for the nth term of the sequence. If the sequence is arithmetic, use the formula:

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

If the sequence is geometric, use the formula:

$$a_n = a_1 \cdot r^{n-1}$$

Q: What if I don't know the common difference or ratio of the sequence?

A: If you don't know the common difference or ratio of the sequence, you can try to find it by looking at the differences or ratios between consecutive terms. Alternatively, you can use a calculator or computer program to find the 100th term of the sequence.

Q: Can I find the nth term of a sequence if I only know the first few terms?

A: Yes, you can find the nth term of a sequence if you only know the first few terms. To do this, you can use the formula for the nth term of the sequence and plug in the values of the first few terms.

Q: What if the sequence is not arithmetic or geometric?

A: If the sequence is not arithmetic or geometric, you may need to use a different formula or method to find the 100th term. For example, if the sequence is a quadratic sequence, you can use the formula:

$$a_n = an^2 + bn + c$$

where $a$, $b$, and $c$ are constants.

Q: Can I use a calculator or computer program to find the 100th term of a sequence?

A: Yes, you can use a calculator or computer program to find the 100th term of a sequence. Many calculators and computer programs have built-in functions for finding the nth term of a sequence.

Key Takeaways

• To determine whether a sequence is arithmetic or geometric, look at the difference or ratio between consecutive terms.
• The formula for finding the nth term of an arithmetic sequence is: $a_n = a_1 + (n-1)d$
• The formula for finding the nth term of a geometric sequence is: $a_n = a_1 \cdot r^{n-1}$
• To find the 100th term of a sequence, use the formula for the nth term of the sequence.
• If you don't know the common difference or ratio of the sequence, try to find it by looking at the differences or ratios between consecutive terms.

Practice Problems

1. Find the 100th term of the sequence: $2, 6, 10, 14, \ldots$
2. Find the 100th term of the sequence: $3, 9, 27, 81, \ldots$
3. Find the nth term of the sequence: $4, 8, 12, 16, \ldots$
4. Find the nth term of the sequence: $5, 25, 125, 625, \ldots$

Solutions

1. $a_{100} = 2 + (100-1)4 = 2 + 396 = 398$
2. $a_{100} = 3 \cdot 3^{100-1} = 3^{99}$
3. $a_n = 4 + (n-1)4 = 4 + 4n - 4 = 4n$
4. $a_n = 5 \cdot 5^{n-1} = 5^n$