C. The 100th Term Of The Sequence $1, 3, 9, \ldots$ Is $3^{99}$.(Simplify Your Answer. Type Your Answer Using Exponential Notation.)The $n$th Term Of The Sequence $1, 3, 9, \ldots$ Is
Introduction
In mathematics, a geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. The nth term of a geometric sequence can be found using a simple formula. In this article, we will explore the formula for the nth term of a geometric sequence and apply it to a specific sequence to find the 100th term.
What is a Geometric Sequence?
A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. The common ratio is denoted by the letter 'r'. For example, the sequence 1, 3, 9, 27, ... is a geometric sequence with a common ratio of 3.
The Formula for the nth Term of a Geometric Sequence
The formula for the nth term of a geometric sequence is given by:
an = ar^(n-1)
where:
- an is the nth term of the sequence
- a is the first term of the sequence
- r is the common ratio
- n is the term number
Applying the Formula to the Given Sequence
In the given sequence 1, 3, 9, ..., the first term is 1 and the common ratio is 3. We are asked to find the 100th term of the sequence. Using the formula for the nth term of a geometric sequence, we can plug in the values as follows:
a = 1 r = 3 n = 100
an = ar^(n-1) = 1 * 3^(100-1) = 3^99
Therefore, the 100th term of the sequence is 3^99.
Conclusion
In this article, we explored the formula for the nth term of a geometric sequence and applied it to a specific sequence to find the 100th term. We found that the 100th term of the sequence 1, 3, 9, ... is 3^99. This formula can be used to find the nth term of any geometric sequence, given the first term and the common ratio.
Example Problems
- Find the 50th term of the sequence 2, 6, 18, ...
- Find the 25th term of the sequence 3, 9, 27, ...
- Find the 100th term of the sequence 4, 16, 64, ...
Solutions
- The 50th term of the sequence 2, 6, 18, ... is 2 * 3^(50-1) = 3^49.
- The 25th term of the sequence 3, 9, 27, ... is 3 * 3^(25-1) = 3^24.
- The 100th term of the sequence 4, 16, 64, ... is 4 * 4^(100-1) = 4^99.
Practice Problems
- Find the nth term of the sequence 1, 2, 4, ... given that the common ratio is 2.
- Find the nth term of the sequence 3, 9, 27, ... given that the common ratio is 3.
- Find the nth term of the sequence 2, 6, 18, ... given that the common ratio is 3.
Solutions
- The nth term of the sequence 1, 2, 4, ... is 1 * 2^(n-1) = 2^(n-1).
- The nth term of the sequence 3, 9, 27, ... is 3 * 3^(n-1) = 3^n.
- The nth term of the sequence 2, 6, 18, ... is 2 * 3^(n-1) = 2 * 3^(n-1).
Q&A: The nth Term of a Geometric Sequence =============================================
Q: What is a geometric sequence?
A: A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.
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 given by:
an = ar^(n-1)
where:
- an is the nth term of the sequence
- a is the first term of the sequence
- r is the common ratio
- 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 need to know the first term and the common ratio. You can then plug these values into the formula for the nth term of a geometric sequence.
Q: What if I don't know the common ratio?
A: If you don't know the common ratio, you can try to find it by looking at the pattern of the sequence. For example, if the sequence is 1, 3, 9, 27, ..., you can see that each term is obtained by multiplying the previous term by 3. Therefore, the common ratio is 3.
Q: Can I use the formula for the nth term of a geometric sequence to find the first term?
A: No, the formula for the nth term of a geometric sequence is used to find the nth term given the first term and the common ratio. If you know the nth term and the common ratio, you can use the formula to find the first term.
Q: What if the common ratio is not an integer?
A: If the common ratio is not an integer, you can still use the formula for the nth term of a geometric sequence. However, you will need to use a calculator or computer to evaluate the expression.
Q: Can I use the formula for the nth term of a geometric sequence to find the sum of a geometric sequence?
A: No, the formula for the nth term of a geometric sequence is used to find the nth term, not the sum of the sequence. To find the sum of a geometric sequence, you will need to use a different formula.
Q: What is the sum of a geometric sequence?
A: The sum of a geometric sequence is given by the formula:
S = a * (1 - r^n) / (1 - r)
where:
- S is the sum of the sequence
- a is the first term of the sequence
- r is the common ratio
- n is the number of terms
Q: Can I use the formula for the nth term of a geometric sequence to find the product of a geometric sequence?
A: No, the formula for the nth term of a geometric sequence is used to find the nth term, not the product of the sequence. To find the product of a geometric sequence, you will need to use a different formula.
Q: What is the product of a geometric sequence?
A: The product of a geometric sequence is given by the formula:
P = a * r^(n-1)
where:
- P is the product of the sequence
- a is the first term of the sequence
- r is the common ratio
- n is the number of terms
Q: Can I use the formula for the nth term of a geometric sequence to find the average of a geometric sequence?
A: No, the formula for the nth term of a geometric sequence is used to find the nth term, not the average of the sequence. To find the average of a geometric sequence, you will need to use a different formula.
Q: What is the average of a geometric sequence?
A: The average of a geometric sequence is given by the formula:
A = (a + ar + ar^2 + ... + ar^(n-1)) / n
where:
- A is the average of the sequence
- a is the first term of the sequence
- r is the common ratio
- n is the number of terms
Q: Can I use the formula for the nth term of a geometric sequence to find the median of a geometric sequence?
A: No, the formula for the nth term of a geometric sequence is used to find the nth term, not the median of the sequence. To find the median of a geometric sequence, you will need to use a different formula.
Q: What is the median of a geometric sequence?
A: The median of a geometric sequence is given by the formula:
M = (ar^(n/2) + ar^(n/2-1) + ... + ar^(n/2+1)) / n
where:
- M is the median of the sequence
- a is the first term of the sequence
- r is the common ratio
- n is the number of terms