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 sequence defined by a recursive formula. The sequence is given by the formula $a_n = -\frac{3}{4} \cdot a_{n-1}$, where each term is obtained by multiplying the previous term by $-\frac{3}{4}$. We are also given the third term of the sequence, $a_3 = \frac{9}{16}$. Our goal is to find the seventh term of the sequence.

Understanding the Recursive Formula

The recursive formula $a_n = -\frac{3}{4} \cdot a_{n-1}$ indicates that each term of the sequence is obtained by multiplying the previous term by $-\frac{3}{4}$. This means that the sequence will exhibit a pattern of multiplication by $-\frac{3}{4}$ as we move from one term to the next.

Finding the Fourth Term

To find the fourth term, we can use the recursive formula and the given value of the third term.

a4=−34⋅a3a_4 = -\frac{3}{4} \cdot a_3

Substituting the value of $a_3 = \frac{9}{16}$, we get:

a4=−34⋅916a_4 = -\frac{3}{4} \cdot \frac{9}{16}

Simplifying the expression, we get:

a4=−2764a_4 = -\frac{27}{64}

Finding the Fifth Term

To find the fifth term, we can use the recursive formula and the value of the fourth term.

a5=−34⋅a4a_5 = -\frac{3}{4} \cdot a_4

Substituting the value of $a_4 = -\frac{27}{64}$, we get:

a5=−34⋅−2764a_5 = -\frac{3}{4} \cdot -\frac{27}{64}

Simplifying the expression, we get:

a5=81256a_5 = \frac{81}{256}

Finding the Sixth Term

To find the sixth term, we can use the recursive formula and the value of the fifth term.

a6=−34⋅a5a_6 = -\frac{3}{4} \cdot a_5

Substituting the value of $a_5 = \frac{81}{256}$, we get:

a6=−34⋅81256a_6 = -\frac{3}{4} \cdot \frac{81}{256}

Simplifying the expression, we get:

a6=−2431024a_6 = -\frac{243}{1024}

Finding the Seventh Term

To find the seventh term, we can use the recursive formula and the value of the sixth term.

a7=−34⋅a6a_7 = -\frac{3}{4} \cdot a_6

Substituting the value of $a_6 = -\frac{243}{1024}$, we get:

a7=−34⋅−2431024a_7 = -\frac{3}{4} \cdot -\frac{243}{1024}

Simplifying the expression, we get:

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

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Q&A: Understanding the Sequence

Q: What is the recursive formula for the sequence?

A: The recursive formula for the sequence is $a_n = -\frac{3}{4} \cdot a_{n-1}$, where each term is obtained by multiplying the previous term by $-\frac{3}{4}$.

Q: What is the given value of the third term?

A: The given value of the third term is $a_3 = \frac{9}{16}$.

Q: How do we find the fourth term?

A: To find the fourth term, we can use the recursive formula and the given value of the third term.

a4=−34⋅a3a_4 = -\frac{3}{4} \cdot a_3

Substituting the value of $a_3 = \frac{9}{16}$, we get:

a4=−34⋅916a_4 = -\frac{3}{4} \cdot \frac{9}{16}

Simplifying the expression, we get:

a4=−2764a_4 = -\frac{27}{64}

Q: How do we find the fifth term?

A: To find the fifth term, we can use the recursive formula and the value of the fourth term.

a5=−34⋅a4a_5 = -\frac{3}{4} \cdot a_4

Substituting the value of $a_4 = -\frac{27}{64}$, we get:

a5=−34⋅−2764a_5 = -\frac{3}{4} \cdot -\frac{27}{64}

Simplifying the expression, we get:

a5=81256a_5 = \frac{81}{256}

Q: How do we find the sixth term?

A: To find the sixth term, we can use the recursive formula and the value of the fifth term.

a6=−34⋅a5a_6 = -\frac{3}{4} \cdot a_5

Substituting the value of $a_5 = \frac{81}{256}$, we get:

a6=−34⋅81256a_6 = -\frac{3}{4} \cdot \frac{81}{256}

Simplifying the expression, we get:

a6=−2431024a_6 = -\frac{243}{1024}

Q: How do we find the seventh term?

A: To find the seventh term, we can use the recursive formula and the value of the sixth term.

a7=−34⋅a6a_7 = -\frac{3}{4} \cdot a_6

Substituting the value of $a_6 = -\frac{243}{1024}$, we get:

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