Question2)
Consider the sequence $1,-\frac{2}{3}, \frac{4}{9},-\frac{8}{27}, \ldots$. Determine $s_{8}$.

Studdy Solution
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:
$$\boxed{-\frac{128}{2187}}$$