1. **State the problem:** Solve for the expression: $$104.5 + 4 \cdot \frac{9}{12} = \left(4 \cdot a_{575} + \frac{100}{(1 + r)^5}\right) \cdot (1 + r)^{\frac{9}{12}}$$ where $$r = 0.028578$$ and $$a_{575} = \frac{1 - (1 + r)^{-5}}{r}$$ 2. **Calculate the left side:** Calculate the multiplication: $$4 \cdot \frac{9}{12} = 4 \cdot 0.75 = 3$$ Add to 104.5: $$104.5 + 3 = 107.5$$ 3. **Calculate $a_{575}$:** Use the formula: $$a_{575} = \frac{1 - (1 + r)^{-5}}{r}$$ Calculate $(1 + r)$: $$1 + 0.028578 = 1.028578$$ Calculate $(1 + r)^{-5}$: $$1.028578^{-5} = \frac{1}{1.028578^5}$$ Calculate $1.028578^5$: $$1.028578^5 \approx 1.1509$$ So, $$1.028578^{-5} \approx \frac{1}{1.1509} = 0.8690$$ Calculate numerator: $$1 - 0.8690 = 0.1310$$ Divide by $r$: $$a_{575} = \frac{0.1310}{0.028578} \approx 4.583$$ 4. **Calculate the right side inside the parentheses:** Calculate $4 \cdot a_{575}$: $$4 \cdot 4.583 = 18.332$$ Calculate denominator term: $$(1 + r)^5 = 1.1509$$ Calculate fraction: $$\frac{100}{1.1509} \approx 86.89$$ Sum inside parentheses: $$18.332 + 86.89 = 105.222$$ 5. **Calculate the multiplier $(1 + r)^{9/12}$:** Calculate exponent: $$\frac{9}{12} = 0.75$$ Calculate: $$(1.028578)^{0.75} \approx e^{0.75 \ln(1.028578)}$$ Calculate $\ln(1.028578) \approx 0.0282$: So, $$e^{0.75 \times 0.0282} = e^{0.02115} \approx 1.0214$$ 6. **Calculate the right side total:** Multiply: $$105.222 \times 1.0214 \approx 107.45$$ 7. **Compare both sides:** Left side = 107.5 Right side = 107.45 They are approximately equal, confirming the expression. **Final answer:** $$107.5 \approx 107.45$$ The equation holds true with given values.