Question

Given the geometric sequence 512 over 243 comma negative 128 over 81 comma 32 comma 27 comma and continuing comma what is a6?

Answer 1

Given the geometric sequence:

$\frac{512}{243},-\frac{128}{81},\frac{32}{27}$

Let's find the 6th term of the sequence a6.

Apply the geometric sequence formula:

$a_n=a_1(r)^{n-1}$

Where:

a1 is the first term = 512/243

n is the number of terms = 6

r is the common ratio.

To find the common ratio, the a term by the term preceding it.

Thus, we have:

$r=-\frac{128}{81}\div\frac{512}{243}$

To divide flip the fraction on the right and change the division symbol to multiplication:

$\begin{gathered} r=-\frac{128}{81}\ast\frac{243}{512} \\ \\ r=-\frac{3}{4} \end{gathered}$

The common ratio is -3/4

Thus, to find the 6th term, a6, we have:

$\begin{gathered} a_6=\frac{512}{243}(-\frac{3}{4})^{6-1} \\ \\ a_6=\frac{512}{243}(-\frac{3}{4})^5 \\ \\ a_6=\frac{512}{243}(-\frac{3^5}{4^5}) \\ \\ a_6=\frac{512}{243}\ast(-\frac{243}{1024}) \end{gathered}$

Solving further:

$\begin{gathered} a_6=-\frac{512}{1024} \\ \\ a_6=-\frac{1}{2} \end{gathered}$

Therefore, the sixth term, a6 is

$-\frac{1}{2}$

ANSWER:

$-\frac{1}{2}$