Evaluate: \text{begin}array l 8,

Expression: $\begin{array} { l }8,& -\frac{ 32 }{ 3 },& \frac{ 128 }{ 9 },& -\frac{ 512 }{ 27 },& …\end{array}$

Calculate the ratio between each pair of consecutive terms

$\begin{array} { l }r=\frac{ -\frac{ 32 }{ 3 } }{ 8 },\\r=\frac{ \frac{ 128 }{ 9 } }{ -\frac{ 32 }{ 3 } },\\r=\frac{ -\frac{ 512 }{ 27 } }{ \frac{ 128 }{ 9 } }\end{array}$

Simplify the expression

$\begin{array} { l }r=-\frac{ 4 }{ 3 },\\r=\frac{ \frac{ 128 }{ 9 } }{ -\frac{ 32 }{ 3 } },\\r=\frac{ -\frac{ 512 }{ 27 } }{ \frac{ 128 }{ 9 } }\end{array}$

Simplify the expression

$\begin{array} { l }r=-\frac{ 4 }{ 3 },\\r=-\frac{ 4 }{ 3 },\\r=\frac{ -\frac{ 512 }{ 27 } }{ \frac{ 128 }{ 9 } }\end{array}$

Simplify the expression

$\begin{array} { l }r=-\frac{ 4 }{ 3 },\\r=-\frac{ 4 }{ 3 },\\r=-\frac{ 4 }{ 3 }\end{array}$

Since the ratio between each pair of consecutive terms is the same, the sequence is geometric and the common ratio is $r=-\frac{ 4 }{ 3 }$

$r=-\frac{ 4 }{ 3 }$

To find the next term, multiply the last term $-\frac{ 512 }{ 27 }$ by the common ratio $r=-\frac{ 4 }{ 3 }$

$-\frac{ 512 }{ 27 } \times \left( -\frac{ 4 }{ 3 } \right)$

Calculate the product

$\frac{ 2048 }{ 81 }$

To find the next term, multiply the last term $\frac{ 2048 }{ 81 }$ by the common ratio $r=-\frac{ 4 }{ 3 }$

$\frac{ 2048 }{ 81 } \times \left( -\frac{ 4 }{ 3 } \right)$

Calculate the product

$-\frac{ 8192 }{ 243 }$

To find the next term, multiply the last term $-\frac{ 8192 }{ 243 }$ by the common ratio $r=-\frac{ 4 }{ 3 }$

$-\frac{ 8192 }{ 243 } \times \left( -\frac{ 4 }{ 3 } \right)$

Calculate the product

$\frac{ 32768 }{ 729 }$

To find the next term, multiply the last term $\frac{ 32768 }{ 729 }$ by the common ratio $r=-\frac{ 4 }{ 3 }$

$\frac{ 32768 }{ 729 } \times \left( -\frac{ 4 }{ 3 } \right)$

Calculate the product

$-\frac{ 131072 }{ 2187 }$

The next four terms are $\begin{array} { l }\frac{ 2048 }{ 81 },& -\frac{ 8192 }{ 243 },& \frac{ 32768 }{ 729 },& -\frac{ 131072 }{ 2187 }\end{array}$

$\begin{array} { l }\frac{ 2048 }{ 81 },& -\frac{ 8192 }{ 243 },& \frac{ 32768 }{ 729 },& -\frac{ 131072 }{ 2187 }\end{array}$