A Sequence Is Defined By A N = − 3 4 A N − 1 A_n = -\frac{3}{4} A_{n-1} A N ​ = − 4 3 ​ A N − 1 ​ , And A 3 = 9 16 A_3 = \frac{9}{16} A 3 ​ = 16 9 ​ . What Is The Seventh Term?A. − 225 256 -\frac{225}{256} − 256 225 ​ B. − 27 64 -\frac{27}{64} − 64 27 ​ C. 59 , 049 1 , 048 , 576 \frac{59,049}{1,048,576} 1 , 048 , 576 59 , 049 ​ D.

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Introduction

In this article, we will explore a recursive sequence defined by the formula an=−34an−1a_n = -\frac{3}{4} a_{n-1}, where a3=916a_3 = \frac{9}{16}. Our goal is to find the seventh term of this sequence. To achieve this, we will use the given recursive formula and the initial condition to calculate each term step by step.

Understanding the Recursive Formula

The recursive formula an=−34an−1a_n = -\frac{3}{4} a_{n-1} indicates that each term in the sequence is obtained by multiplying the previous term by −34-\frac{3}{4}. This means that the sign of each term will alternate between positive and negative, and the magnitude of each term will decrease by a factor of 34\frac{3}{4}.

Calculating the Fourth Term

To find the fourth term, we can use the recursive formula and the given initial condition a3=916a_3 = \frac{9}{16}. We have:

a4=−34a3=−34⋅916=−2764a_4 = -\frac{3}{4} a_3 = -\frac{3}{4} \cdot \frac{9}{16} = -\frac{27}{64}

Calculating the Fifth Term

Using the recursive formula and the value of the fourth term, we can calculate the fifth term:

a5=−34a4=−34⋅−2764=81256a_5 = -\frac{3}{4} a_4 = -\frac{3}{4} \cdot -\frac{27}{64} = \frac{81}{256}

Calculating the Sixth Term

Now, we can use the recursive formula and the value of the fifth term to calculate the sixth term:

a6=−34a5=−34⋅81256=−2431024a_6 = -\frac{3}{4} a_5 = -\frac{3}{4} \cdot \frac{81}{256} = -\frac{243}{1024}

Calculating the Seventh Term

Finally, we can use the recursive formula and the value of the sixth term to calculate the seventh term:

a7=−34a6=−34⋅−2431024=7294096a_7 = -\frac{3}{4} a_6 = -\frac{3}{4} \cdot -\frac{243}{1024} = \frac{729}{4096}

However, we can simplify this fraction further by dividing both the numerator and the denominator by their greatest common divisor, which is 1.

Conclusion

In this article, we have used the recursive formula an=−34an−1a_n = -\frac{3}{4} a_{n-1} and the initial condition a3=916a_3 = \frac{9}{16} to find the seventh term of the sequence. We have calculated each term step by step, using the recursive formula to obtain the value of each term. The seventh term of the sequence is 7294096\frac{729}{4096}.

Answer

Introduction

In our previous article, we explored a recursive sequence defined by the formula an=−34an−1a_n = -\frac{3}{4} a_{n-1}, where a3=916a_3 = \frac{9}{16}. We used the recursive formula and the initial condition to calculate each term step by step and found the seventh term of the sequence. In this article, we will answer some frequently asked questions related to this sequence.

Q: What is the general formula for the nth term of this sequence?

A: The general formula for the nth term of this sequence is an=(−34)n−3⋅916a_n = \left(-\frac{3}{4}\right)^{n-3} \cdot \frac{9}{16}.

Q: How do I calculate the nth term of this sequence?

A: To calculate the nth term of this sequence, you can use the recursive formula an=−34an−1a_n = -\frac{3}{4} a_{n-1} and the initial condition a3=916a_3 = \frac{9}{16}. Alternatively, you can use the general formula an=(−34)n−3⋅916a_n = \left(-\frac{3}{4}\right)^{n-3} \cdot \frac{9}{16}.

Q: What is the pattern of the sequence?

A: The pattern of the sequence is that each term is obtained by multiplying the previous term by −34-\frac{3}{4}. This means that the sign of each term will alternate between positive and negative, and the magnitude of each term will decrease by a factor of 34\frac{3}{4}.

Q: How do I determine the sign of each term in the sequence?

A: To determine the sign of each term in the sequence, you can use the fact that the sign of each term alternates between positive and negative. If the previous term is positive, the next term will be negative, and vice versa.

Q: Can I use this sequence in real-world applications?

A: Yes, you can use this sequence in real-world applications such as modeling population growth or decline, predicting stock prices, or analyzing the behavior of complex systems.

Q: How do I extend this sequence to more terms?

A: To extend this sequence to more terms, you can continue to use the recursive formula an=−34an−1a_n = -\frac{3}{4} a_{n-1} and the initial condition a3=916a_3 = \frac{9}{16}. Alternatively, you can use the general formula an=(−34)n−3⋅916a_n = \left(-\frac{3}{4}\right)^{n-3} \cdot \frac{9}{16}.

Q: Can I modify this sequence to create a new sequence?

A: Yes, you can modify this sequence to create a new sequence by changing the recursive formula or the initial condition. For example, you can change the recursive formula to an=34an−1a_n = \frac{3}{4} a_{n-1} or change the initial condition to a3=−916a_3 = -\frac{9}{16}.

Conclusion

In this article, we have answered some frequently asked questions related to the recursive sequence defined by the formula an=−34an−1a_n = -\frac{3}{4} a_{n-1}, where a3=916a_3 = \frac{9}{16}. We have provided the general formula for the nth term of this sequence, explained how to calculate the nth term, and discussed the pattern of the sequence. We have also explored the possibility of using this sequence in real-world applications and extending it to more terms.