1. **State the problem:** Simplify the expression $$\frac{50\left(1-0.8^{8-1}\right)}{\left(1-0.8\right)}$$. 2. **Identify the formula:** This expression resembles the sum of a geometric series formula $$S_n = a \frac{1-r^n}{1-r}$$ where $a=50$, $r=0.8$, and $n=8-1=7$. 3. **Calculate the power:** Compute $0.8^{7}$. $$0.8^{7} = 0.2097152$$ 4. **Substitute the value:** $$1 - 0.8^{7} = 1 - 0.2097152 = 0.7902848$$ 5. **Calculate the denominator:** $$1 - 0.8 = 0.2$$ 6. **Substitute all values back into the expression:** $$\frac{50 \times 0.7902848}{0.2}$$ 7. **Multiply numerator:** $$50 \times 0.7902848 = 39.51424$$ 8. **Divide by denominator:** $$\frac{39.51424}{0.2} = 197.5712$$ **Final answer:** $$197.5712$$