A Sequence Is Defined By $a_n = -\frac{3}{4} \cdot A_{n-1}$, And $a_3 = \frac{9}{16}$. 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 $a_n = -\frac{3}{4} \cdot a_{n-1}$, where $a_3 = \frac{9}{16}$. Our goal is to find the seventh term of this sequence. To do this, we will use the given formula to calculate each term, starting from the third term, and then use the result to find the seventh term.

Understanding the Recursive Formula

The recursive formula $a_n = -\frac{3}{4} \cdot a_{n-1}$ indicates that each term in the sequence is obtained by multiplying the previous term by $-\frac{3}{4}$. 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 $\frac{3}{4}$.

Calculating the Fourth Term

To find the fourth term, we will use the recursive formula with $a_3 = \frac{9}{16}$.

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

Calculating the Fifth Term

To find the fifth term, we will use the recursive formula with $a_4 = -\frac{27}{64}$.

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

Calculating the Sixth Term

To find the sixth term, we will use the recursive formula with $a_5 = \frac{81}{256}$.

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

Calculating the Seventh Term

To find the seventh term, we will use the recursive formula with $a_6 = -\frac{243}{1024}$.

a7=−34⋅a6=−34⋅−2431024=7294096a_7 = -\frac{3}{4} \cdot 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.

a7=7294096a_7 = \frac{729}{4096}

But we can simplify this fraction further by dividing both the numerator and the denominator by 9.

a7=81512a_7 = \frac{81}{512}

But we can simplify this fraction further by dividing both the numerator and the denominator by 81.

a7=164a_7 = \frac{1}{64}

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Q: What is the recursive formula for the sequence?

A: The recursive formula for the sequence is $a_n = -\frac{3}{4} \cdot a_{n-1}$.

Q: What is the initial term of the sequence?

A: The initial term of the sequence is $a_3 = \frac{9}{16}$.

Q: How do we find the subsequent terms of the sequence?

A: We find the subsequent terms of the sequence by using the recursive formula with the previous term. For example, to find the fourth term, we use the recursive formula with the third term: $a_4 = -\frac{3}{4} \cdot a_3$.

Q: Can we simplify the fraction $\frac{729}{4096}$ further?

A: Yes, we can simplify the fraction $\frac{729}{4096}$ further by dividing both the numerator and the denominator by their greatest common divisor, which is 1.

Q: What is the simplified fraction of $\frac{729}{4096}$?

A: The simplified fraction of $\frac{729}{4096}$ is $\frac{81}{512}$.

Q: Can we simplify the fraction $\frac{81}{512}$ further?

A: Yes, we can simplify the fraction $\frac{81}{512}$ further by dividing both the numerator and the denominator by their greatest common divisor, which is 1.

Q: What is the simplified fraction of $\frac{81}{512}$?

A: The simplified fraction of $\frac{81}{512}$ is $\frac{1}{64}$.

Q: Is the fraction $\frac{1}{64}$ the final answer?

A: No, we can simplify the fraction $\frac{1}{64}$ further by dividing both the numerator and the denominator by their greatest common divisor, which is 1.

Q: What is the simplified fraction of $\frac{1}{64}$?

A: The simplified fraction of $\frac{1}{64}$ is $\frac{1}{64}$.

Q: Why is the fraction $\frac{1}{64}$ not the final answer?

A: The fraction $\frac{1}{64}$ is not the final answer because we can simplify it further by dividing both the numerator and the denominator by their greatest common divisor, which is 1.

Q: What is the final answer?

A: The final answer is $\frac{1}{64}$.

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a_7 = \frac{1