A Geometric Sequence Starts With A First Term Of 4 And Has A Common Ratio Of 7.Write The Recursive Formula Of This Sequence.
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 recursive formula of a geometric sequence with a first term of 4 and a common ratio of 7.
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 general form of a geometric sequence is:
a, ar, ar^2, ar^3, ...
where a is the first term and r is the common ratio.
Recursive Formula of a Geometric Sequence
The recursive formula 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, r is the common ratio, and n is the term number.
Example: Geometric Sequence with First Term 4 and Common Ratio 7
Let's consider a geometric sequence with a first term of 4 and a common ratio of 7. We can use the recursive formula to find the nth term of the sequence.
an = 4(7)^(n-1)
Finding the nth Term
To find the nth term of the sequence, we can plug in the value of n into the recursive formula.
For example, to find the 5th term of the sequence, we can plug in n = 5:
a5 = 4(7)^(5-1) = 4(7)^4 = 4(2401) = 9604
Properties of Geometric Sequences
Geometric sequences have several important properties that make them useful in mathematics and other fields. Some of the key properties of geometric sequences include:
- Common Ratio: The common ratio of a geometric sequence is the fixed number that is multiplied by each term to get the next term.
- First Term: The first term of a geometric sequence is the first number in the sequence.
- Term Number: The term number of a geometric sequence is the position of the term in the sequence.
- Recursive Formula: The recursive formula of a geometric sequence is a formula that describes how each term is related to the previous term.
Solving Geometric Sequences
Geometric sequences can be solved using various methods, including:
- Recursive Formula: The recursive formula can be used to find the nth term of the sequence.
- Closed-Form Formula: A closed-form formula can be used to find the nth term of the sequence without having to use the recursive formula.
- Arithmetic Series: An arithmetic series can be used to find the sum of the first n terms of the sequence.
Applications of Geometric Sequences
Geometric sequences have many applications in mathematics and other fields, including:
- Finance: Geometric sequences are used to calculate interest rates and investment returns.
- Biology: Geometric sequences are used to model population growth and decline.
- Computer Science: Geometric sequences are used to model the growth of algorithms and data structures.
Conclusion
In conclusion, geometric sequences are an important type of sequence that have many applications in mathematics and other fields. The recursive formula of a geometric sequence is a powerful tool for finding the nth term of the sequence, and it has many properties that make it useful in various contexts. By understanding the properties and applications of geometric sequences, we can better appreciate the beauty and power of mathematics.
References
- Krantz, S. G. (2013). Calculus: Early Transcendentals. 2nd ed. Pearson Education.
- Larson, R. E. (2013). Calculus: An Applied Approach. 8th ed. Cengage Learning.
- Stewart, J. (2015). Calculus: Early Transcendentals. 8th ed. Cengage Learning.
A Geometric Sequence: Recursive Formula and Properties ===========================================================
Q&A: Geometric Sequences
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 recursive formula of a geometric sequence?
A: The recursive formula 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, 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 recursive formula:
an = ar^(n-1)
For example, to find the 5th term of a geometric sequence with a first term of 4 and a common ratio of 7, you can plug in n = 5:
a5 = 4(7)^(5-1) = 4(7)^4 = 4(2401) = 9604
Q: What are some common applications of geometric sequences?
A: Geometric sequences have many applications in mathematics and other fields, including:
- Finance: Geometric sequences are used to calculate interest rates and investment returns.
- Biology: Geometric sequences are used to model population growth and decline.
- Computer Science: Geometric sequences are used to model the growth of algorithms and data structures.
Q: How do I solve a geometric sequence?
A: Geometric sequences can be solved using various methods, including:
- Recursive Formula: The recursive formula can be used to find the nth term of the sequence.
- Closed-Form Formula: A closed-form formula can be used to find the nth term of the sequence without having to use the recursive formula.
- Arithmetic Series: An arithmetic series can be used to find the sum of the first n terms of the sequence.
Q: What are some common properties of geometric sequences?
A: Geometric sequences have several important properties that make them useful in mathematics and other fields, including:
- Common Ratio: The common ratio of a geometric sequence is the fixed number that is multiplied by each term to get the next term.
- First Term: The first term of a geometric sequence is the first number in the sequence.
- Term Number: The term number of a geometric sequence is the position of the term in the sequence.
- Recursive Formula: The recursive formula of a geometric sequence is a formula that describes how each term is related to the previous term.
Q: Can I use geometric sequences to model real-world phenomena?
A: Yes, geometric sequences can be used to model many real-world phenomena, including population growth, investment returns, and algorithm growth.
Q: How do I determine if a sequence is geometric?
A: To determine if a sequence is geometric, you can check if each term is found by multiplying the previous term by a fixed number. If this is the case, then the sequence is geometric.
Q: What are some common mistakes to avoid when working with geometric sequences?
A: Some common mistakes to avoid when working with geometric sequences include:
- Forgetting to multiply by the common ratio: Make sure to multiply each term by the common ratio to get the next term.
- Using the wrong formula: Make sure to use the correct formula for the geometric sequence, such as the recursive formula or the closed-form formula.
- Not checking for convergence: Make sure to check if the sequence converges to a limit before using it to model real-world phenomena.
Conclusion
In conclusion, geometric sequences are an important type of sequence that have many applications in mathematics and other fields. By understanding the properties and applications of geometric sequences, we can better appreciate the beauty and power of mathematics.