Math Snap

STEP 1

1. The sequence is geometric.
2. The first term of the sequence is $a = 1$.
3. The common ratio $r$ can be determined from the sequence terms.
4. The formula for the $n$-th term of a geometric sequence is $s_n = a \cdot r^{n-1}$.

STEP 2

1. Identify the common ratio $r$.
2. Use the formula for the $n$-th term of a geometric sequence to find $s_8$.

STEP 3

Identify the common ratio $r$ by dividing the second term by the first term:
$$r = \frac{-\frac{2}{3}}{1} = -\frac{2}{3}$$ Verify by checking the ratio between subsequent terms:
$$r = \frac{\frac{4}{9}}{-\frac{2}{3}} = -\frac{2}{3}$$ $$r = \frac{-\frac{8}{27}}{\frac{4}{9}} = -\frac{2}{3}$$ The common ratio $r$ is consistent.

Use the formula for the $n$-th term of a geometric sequence to find $s_8$:
The formula is:
$$s_n = a \cdot r^{n-1}$$ Substitute $a = 1$, $r = -\frac{2}{3}$, and $n = 8$:
$$s_8 = 1 \cdot \left(-\frac{2}{3}\right)^{8-1}$$ $$s_8 = \left(-\frac{2}{3}\right)^7$$ Calculate $\left(-\frac{2}{3}\right)^7$:
$$\left(-\frac{2}{3}\right)^7 = -\frac{2^7}{3^7} = -\frac{128}{2187}$$ The value of $s_8$ is:
$$\boxed{-\frac{128}{2187}}$$