The Recursive Rule For A Geometric Sequence Is Given:${ A_1 = 2 \quad ; \quad A_n = \frac{1}{3} A_{n-1} } E N T E R T H E E X P L I C I T R U L E F O R T H E S E Q U E N C E : Enter The Explicit Rule For The Sequence: E N T Er T H Ee X Pl I C I T R U L E F Or T H Ese Q U E N Ce : { A_n = \square \}

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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. The recursive rule for a geometric sequence is given by the formula: an=13an−1a_n = \frac{1}{3} a_{n-1}, where ana_n is the nth term of the sequence and an−1a_{n-1} is the (n-1)th term. In this article, we will explore how to find the explicit rule for the sequence using the recursive rule.

Understanding the Recursive Rule

The recursive rule for a geometric sequence is given by the formula: an=13an−1a_n = \frac{1}{3} a_{n-1}. This means that to find the nth term of the sequence, we need to multiply the (n-1)th term by 13\frac{1}{3}. For example, if we want to find the 3rd term of the sequence, we would multiply the 2nd term by 13\frac{1}{3}.

Finding the Explicit Rule

To find the explicit rule for the sequence, we need to find a formula that gives us the nth term of the sequence directly, without having to find the previous terms. We can do this by using the recursive rule and the initial term of the sequence.

Let's start by writing the recursive rule: an=13an−1a_n = \frac{1}{3} a_{n-1}. We can rewrite this as: an=13an−1=13(13an−2)=(13)2an−2=(13)3an−3=…=(13)n−1a1a_n = \frac{1}{3} a_{n-1} = \frac{1}{3} \left( \frac{1}{3} a_{n-2} \right) = \left( \frac{1}{3} \right)^2 a_{n-2} = \left( \frac{1}{3} \right)^3 a_{n-3} = \ldots = \left( \frac{1}{3} \right)^{n-1} a_1.

Deriving the Explicit Rule

Now that we have the recursive rule in terms of the initial term, we can derive the explicit rule by substituting the initial term into the formula. The initial term is given as a1=2a_1 = 2. Substituting this into the formula, we get: an=(13)n−1a1=(13)n−12a_n = \left( \frac{1}{3} \right)^{n-1} a_1 = \left( \frac{1}{3} \right)^{n-1} 2.

Simplifying the Explicit Rule

The explicit rule we derived is: an=(13)n−12a_n = \left( \frac{1}{3} \right)^{n-1} 2. We can simplify this by multiplying the fraction by 2: an=2(13)n−1a_n = 2 \left( \frac{1}{3} \right)^{n-1}.

Conclusion

In this article, we derived the explicit rule for a geometric sequence using the recursive rule. We started by writing the recursive rule and then derived the explicit rule by substituting the initial term into the formula. The explicit rule we derived is: an=2(13)n−1a_n = 2 \left( \frac{1}{3} \right)^{n-1}.

Example Use Case

Suppose we want to find the 5th term of the sequence. We can use the explicit rule to find it: a5=2(13)5−1=2(13)4=2(181)=281a_5 = 2 \left( \frac{1}{3} \right)^{5-1} = 2 \left( \frac{1}{3} \right)^4 = 2 \left( \frac{1}{81} \right) = \frac{2}{81}.

Applications of Geometric Sequences

Geometric sequences have many applications in mathematics and other fields. Some examples include:

  • Finance: Geometric sequences are used to model the growth of investments and the decay of debts.
  • Biology: Geometric sequences are used to model the growth of populations and the spread of diseases.
  • Computer Science: Geometric sequences are used to model the growth of algorithms and the decay of data.

Conclusion

In conclusion, geometric sequences are a powerful tool for modeling real-world phenomena. The recursive rule for a geometric sequence is given by the formula: an=13an−1a_n = \frac{1}{3} a_{n-1}. We can derive the explicit rule for the sequence by substituting the initial term into the formula. The explicit rule we derived is: an=2(13)n−1a_n = 2 \left( \frac{1}{3} \right)^{n-1}. Geometric sequences have many applications in mathematics and other fields, and are an important tool for modeling real-world phenomena.

References

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Introduction

Geometric sequences are a fundamental concept in mathematics, and they have many real-world applications. In this article, we will answer some of the most frequently asked questions about 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 rule for a geometric sequence?

A: The recursive rule for a geometric sequence is given by the formula: an=13an−1a_n = \frac{1}{3} a_{n-1}, where ana_n is the nth term of the sequence and an−1a_{n-1} is the (n-1)th term.

Q: How do I find the explicit rule for a geometric sequence?

A: To find the explicit rule for a geometric sequence, you need to find a formula that gives you the nth term of the sequence directly, without having to find the previous terms. You can do this by using the recursive rule and the initial term of the sequence.

Q: What is the explicit rule for a geometric sequence?

A: The explicit rule for a geometric sequence is given by the formula: an=2(13)n−1a_n = 2 \left( \frac{1}{3} \right)^{n-1}.

Q: How do I use the explicit rule to find the nth term of a geometric sequence?

A: To use the explicit rule to find the nth term of a geometric sequence, you need to substitute the value of n into the formula. For example, to find the 5th term of the sequence, you would substitute n = 5 into the formula: a5=2(13)5−1=2(13)4=2(181)=281a_5 = 2 \left( \frac{1}{3} \right)^{5-1} = 2 \left( \frac{1}{3} \right)^4 = 2 \left( \frac{1}{81} \right) = \frac{2}{81}.

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 model the growth of investments and the decay of debts.
  • Biology: Geometric sequences are used to model the growth of populations and the spread of diseases.
  • Computer Science: Geometric sequences are used to model the growth of algorithms and the decay of data.

Q: How do I determine the common ratio of a geometric sequence?

A: To determine the common ratio of a geometric sequence, you need to examine the ratio of consecutive terms. For example, if the sequence is 2, 6, 18, 54, ... , the common ratio is 3, since each term is obtained by multiplying the previous term by 3.

Q: What is the formula for the sum of a geometric sequence?

A: The formula for the sum of a geometric sequence is given by the formula: Sn=a1(1−rn)1−rS_n = \frac{a_1 \left( 1 - r^n \right)}{1 - r}, where SnS_n is the sum of the first n terms, a1a_1 is the first term, rr is the common ratio, and nn is the number of terms.

Q: How do I use the formula for the sum of a geometric sequence?

A: To use the formula for the sum of a geometric sequence, you need to substitute the values of a1a_1, rr, and nn into the formula. For example, to find the sum of the first 5 terms of the sequence 2, 6, 18, 54, ..., you would substitute a1=2a_1 = 2, r=3r = 3, and n=5n = 5 into the formula: S5=2(1−35)1−3=2(1−243)−2=2(−242)−2=242S_5 = \frac{2 \left( 1 - 3^5 \right)}{1 - 3} = \frac{2 \left( 1 - 243 \right)}{-2} = \frac{2 \left( -242 \right)}{-2} = 242.

Conclusion

In conclusion, geometric sequences are a powerful tool for modeling real-world phenomena. By understanding the recursive rule, explicit rule, and real-world applications of geometric sequences, you can use them to solve a wide range of problems. We hope this Q&A guide has been helpful in answering some of the most frequently asked questions about geometric sequences.

References