Write A Recursive And Explicit Formula For Each Sequence:1. ${$5, 10, 20, 40, \ldots$}$2. ${ 3, 7, 11, 15, \ldots\$}

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Introduction

In mathematics, sequences are an essential concept used to describe a list of numbers that follow a specific pattern. Recursive and explicit formulas are two ways to represent sequences. A recursive formula defines each term of a sequence as a function of the preceding term(s), whereas an explicit formula gives a direct expression for each term of the sequence. In this article, we will explore the recursive and explicit formulas for two given sequences.

Sequence 1: 5, 10, 20, 40, ...

Recursive Formula

The given sequence is 5, 10, 20, 40, ... . To find the recursive formula, we need to identify the pattern. Looking at the sequence, we can see that each term is obtained by multiplying the previous term by 2.

a_n = 2a_{n-1}

where a_n is the nth term of the sequence.

Initial Condition

To start the recursion, we need an initial condition. In this case, the first term a_1 is 5.

a_1 = 5

Explicit Formula

To find the explicit formula, we can use the recursive formula and the initial condition. We can rewrite the recursive formula as:

a_n = 2^{n-1}a_1

Substituting the initial condition a_1 = 5, we get:

a_n = 2^{n-1} \cdot 5

This is the explicit formula for the sequence.

Example

Let's find the 5th term of the sequence using the explicit formula.

a_5 = 2^{5-1} \cdot 5
= 2^4 \cdot 5
= 16 \cdot 5
= 80

Sequence 2: 3, 7, 11, 15, ...

Recursive Formula

The given sequence is 3, 7, 11, 15, ... . To find the recursive formula, we need to identify the pattern. Looking at the sequence, we can see that each term is obtained by adding 4 to the previous term.

a_n = a_{n-1} + 4

where a_n is the nth term of the sequence.

Initial Condition

To start the recursion, we need an initial condition. In this case, the first term a_1 is 3.

a_1 = 3

Explicit Formula

To find the explicit formula, we can use the recursive formula and the initial condition. We can rewrite the recursive formula as:

a_n = a_1 + 4(n-1)

Substituting the initial condition a_1 = 3, we get:

a_n = 3 + 4(n-1)

This is the explicit formula for the sequence.

Example

Let's find the 5th term of the sequence using the explicit formula.

a_5 = 3 + 4(5-1)
= 3 + 4 \cdot 4
= 3 + 16
= 19

Conclusion

In this article, we have explored the recursive and explicit formulas for two given sequences. We have seen how to identify the pattern in a sequence and use it to find the recursive formula. We have also seen how to use the recursive formula and the initial condition to find the explicit formula. The explicit formula provides a direct expression for each term of the sequence, making it easier to calculate the terms.

References

  • [1] "Sequences and Series" by Michael Sullivan
  • [2] "Discrete Mathematics" by Kenneth H. Rosen

Further Reading

  • [1] "Recursive Sequences" by Math Open Reference
  • [2] "Explicit Formulas for Sequences" by Wolfram MathWorld
    Recursive and Explicit Formulas for Sequences: Q&A =====================================================

Introduction

In our previous article, we explored the recursive and explicit formulas for two given sequences. In this article, we will answer some frequently asked questions about recursive and explicit formulas for sequences.

Q: What is the difference between a recursive formula and an explicit formula?

A: A recursive formula defines each term of a sequence as a function of the preceding term(s), whereas an explicit formula gives a direct expression for each term of the sequence.

Q: How do I identify the pattern in a sequence to find the recursive formula?

A: To identify the pattern in a sequence, look for a common difference or ratio between consecutive terms. For example, if the sequence is 5, 10, 20, 40, ... , you can see that each term is obtained by multiplying the previous term by 2.

Q: How do I find the explicit formula from the recursive formula?

A: To find the explicit formula from the recursive formula, you can use the initial condition and the recursive formula to rewrite the recursive formula as an explicit formula. For example, if the recursive formula is a_n = 2a_{n-1} and the initial condition is a_1 = 5, you can rewrite the recursive formula as a_n = 2^{n-1}a_1.

Q: What is the initial condition, and why is it important?

A: The initial condition is the first term of the sequence, and it is used to start the recursion. The initial condition is important because it provides the starting point for the recursive formula.

Q: Can I have multiple initial conditions for a recursive formula?

A: No, you can only have one initial condition for a recursive formula. The initial condition is used to start the recursion, and having multiple initial conditions would create ambiguity.

Q: How do I find the nth term of a sequence using the explicit formula?

A: To find the nth term of a sequence using the explicit formula, simply substitute n into the explicit formula and evaluate the expression. For example, if the explicit formula is a_n = 2^{n-1} \cdot 5, you can find the 5th term by substituting n = 5 into the formula.

Q: Can I use the recursive formula to find the nth term of a sequence?

A: Yes, you can use the recursive formula to find the nth term of a sequence. However, this method can be more time-consuming and may not be as efficient as using the explicit formula.

Q: What are some common types of sequences that can be represented using recursive and explicit formulas?

A: Some common types of sequences that can be represented using recursive and explicit formulas include arithmetic sequences, geometric sequences, and Fibonacci sequences.

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

A: To determine whether a sequence is arithmetic or geometric, look for a common difference or ratio between consecutive terms. If the sequence has a common difference, it is an arithmetic sequence. If the sequence has a common ratio, it is a geometric sequence.

Conclusion

In this article, we have answered some frequently asked questions about recursive and explicit formulas for sequences. We have seen how to identify the pattern in a sequence, find the explicit formula from the recursive formula, and use the explicit formula to find the nth term of a sequence.

References

  • [1] "Sequences and Series" by Michael Sullivan
  • [2] "Discrete Mathematics" by Kenneth H. Rosen

Further Reading

  • [1] "Recursive Sequences" by Math Open Reference
  • [2] "Explicit Formulas for Sequences" by Wolfram MathWorld