Enter The Explicit Rule For The Geometric Sequence.${ \begin{array}{l} 120, 40, \frac{40}{3}, \frac{40}{9}, \frac{40}{27}, \ldots \ a_n = 120\left(\frac{1}{3}\right)^{n-1} \end{array} }$
Introduction
A geometric sequence is a type of sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. In this article, we will explore the explicit rule for a geometric sequence, which is a formula that allows us to find any term in the sequence without having to calculate all the preceding terms.
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 number called the common ratio. The common ratio is usually denoted by the letter 'r'. For example, if we have a geometric sequence with the first term 'a' and the common ratio 'r', the sequence will be:
a, ar, ar^2, ar^3, ...
The Explicit Rule for a Geometric Sequence
The explicit rule for a geometric sequence is a formula that allows us to find any term in the sequence without having to calculate all the preceding terms. The formula is given by:
a_n = a * r^(n-1)
where a_n is the nth term of the sequence, a is the first term, r is the common ratio, and n is the term number.
Example: Finding the nth Term of a Geometric Sequence
Let's consider the geometric sequence given in the problem statement:
120, 40, 40/3, 40/9, 40/27, ...
We are given the formula for the nth term of this sequence:
a_n = 120 * (1/3)^(n-1)
To find the 5th term of this sequence, we can plug in n = 5 into the formula:
a_5 = 120 * (1/3)^(5-1) = 120 * (1/3)^4 = 120 * 1/81 = 120/81 = 40/27
Therefore, the 5th term of this sequence is 40/27.
How to Use the Explicit Rule
To use the explicit rule, we need to know the first term 'a' and the common ratio 'r' of the sequence. We can then plug these values into the formula to find any term in the sequence.
For example, let's say we have a geometric sequence with the first term 'a' = 2 and the common ratio 'r' = 3. We can use the explicit rule to find the 4th term of this sequence:
a_4 = 2 * 3^(4-1) = 2 * 3^3 = 2 * 27 = 54
Therefore, the 4th term of this sequence is 54.
Advantages of the Explicit Rule
The explicit rule has several advantages over the recursive formula for a geometric sequence. One of the main advantages is that it allows us to find any term in the sequence without having to calculate all the preceding terms. This makes it much faster and more efficient to work with geometric sequences.
Another advantage of the explicit rule is that it is much easier to understand and work with than the recursive formula. The explicit rule is a simple formula that can be easily applied to any geometric sequence, whereas the recursive formula requires us to understand the concept of a recursive sequence.
Conclusion
In conclusion, the explicit rule for a geometric sequence is a powerful tool that allows us to find any term in the sequence without having to calculate all the preceding terms. The formula is given by:
a_n = a * r^(n-1)
where a_n is the nth term of the sequence, a is the first term, r is the common ratio, and n is the term number.
We have seen how to use the explicit rule to find the nth term of a geometric sequence, and we have discussed the advantages of using the explicit rule over the recursive formula.
Applications of Geometric Sequences
Geometric sequences have many real-world applications, including:
- Finance: Geometric sequences are used to calculate compound interest and to model population growth.
- Biology: Geometric sequences are used to model population growth and to calculate the number of cells in a population.
- Computer Science: Geometric sequences are used to model the growth of algorithms and to calculate the number of operations required to solve a problem.
Common Misconceptions about Geometric Sequences
There are several common misconceptions about geometric sequences that we should be aware of:
- Myth 1: Geometric sequences are only used in mathematics: This is not true. Geometric sequences have many real-world applications in finance, biology, and computer science.
- Myth 2: Geometric sequences are only used to calculate the nth term: This is not true. Geometric sequences can be used to model population growth, calculate compound interest, and to solve many other problems.
Conclusion
Q: What is a geometric sequence?
A: A geometric sequence is a type of sequence 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 common ratio in a geometric sequence?
A: The common ratio is the number that is multiplied by each term to get the next term in the sequence. It is usually denoted by the letter 'r'.
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 explicit rule:
a_n = a * r^(n-1)
where a_n is the nth term of the sequence, 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_n = a * (1 - r^n) / (1 - r)
where S_n is the sum of the first n terms of the sequence, 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 an infinite geometric sequence?
A: To find the sum of an infinite geometric sequence, you can use the formula:
S = a / (1 - r)
where S is the sum of the sequence, a is the first term, and r is the common ratio.
Q: What is the difference between a geometric sequence and an arithmetic sequence?
A: A geometric sequence is a sequence where each term after the first is found by multiplying the previous term by a fixed number, while an arithmetic sequence is a sequence where each term after the first is found by adding a fixed number.
Q: Can I use the explicit rule to find the nth term of an arithmetic sequence?
A: No, the explicit rule is only used for geometric sequences. For arithmetic sequences, you can use the formula:
a_n = a + (n-1)d
where a_n is the nth term of the sequence, a is the first term, n is the term number, and d is the common difference.
Q: How do I determine if a sequence is geometric or arithmetic?
A: To determine if a sequence is geometric or arithmetic, you can look at the ratio between consecutive terms. If the ratio is constant, then the sequence is geometric. If the difference between consecutive terms is constant, then the sequence is arithmetic.
Q: Can I use the explicit rule to find the sum of an arithmetic sequence?
A: No, the explicit rule is only used for geometric sequences. For arithmetic sequences, you can use the formula:
S_n = n/2 * (a + l)
where S_n is the sum of the first n terms of the sequence, n is the number of terms, a is the first term, and l is the last term.
Q: What are some real-world applications of geometric sequences?
A: Geometric sequences have many real-world applications, including:
- Finance: Geometric sequences are used to calculate compound interest and to model population growth.
- Biology: Geometric sequences are used to model population growth and to calculate the number of cells in a population.
- Computer Science: Geometric sequences are used to model the growth of algorithms and to calculate the number of operations required to solve a problem.
Q: Can I use the explicit rule to find the nth term of a sequence that is neither geometric nor arithmetic?
A: No, the explicit rule is only used for geometric sequences. If the sequence is neither geometric nor arithmetic, then you will need to use a different method to find the nth term.
Conclusion
In conclusion, geometric sequences are a powerful tool that can be used to model population growth, calculate compound interest, and to solve many other problems. The explicit rule for a geometric sequence is a simple formula that can be easily applied to any geometric sequence, and it has many advantages over the recursive formula. We have seen how to use the explicit rule to find the nth term of a geometric sequence, and we have discussed the applications and common misconceptions about geometric sequences.