1. **Problem statement:** We have an annuity with payments €650,000, then two payments of amount $X$ each, spaced 52 weeks apart, and a final unknown payment after 3 periods. The discount rate is 11.5% per annum. We want to find $X$ such that the future value (FV) of the annuity after 3 periods is €2,400,000. 2. **Formula and explanation:** The future value of an annuity with payments $P_i$ at times $t_i$ discounted at rate $r$ per annum is $$FV = \sum P_i (1 + r)^{T - t_i}$$ where $T$ is the total time horizon (3 periods here). 3. **Convert discount rate to period rate:** Since periods are 52 weeks (1 year), the period rate is $r = 0.115$. 4. **Set timeline:** - Payment 1: €650,000 at time 0 - Payment 2: $X$ at time 1 - Payment 3: $X$ at time 2 - Payment 4: ??? at time 3 (not needed for $X$ calculation) 5. **Future value at time 3:** $$FV = 650000(1 + 0.115)^3 + X(1 + 0.115)^2 + X(1 + 0.115)^1$$ 6. **Calculate powers:** $$ (1 + 0.115)^3 = 1.115^3 = 1.3861$$ $$ (1 + 0.115)^2 = 1.115^2 = 1.2432$$ $$ (1 + 0.115)^1 = 1.115$$ 7. **Write FV equation:** $$2,400,000 = 650000 \times 1.3861 + X \times 1.2432 + X \times 1.115$$ 8. **Simplify:** $$2,400,000 = 900965 + X(1.2432 + 1.115)$$ $$2,400,000 - 900965 = X \times 2.3582$$ $$1,499,035 = 2.3582 X$$ 9. **Solve for $X$:** $$X = \frac{1,499,035}{2.3582} = 635,800.57$$ **Final answer:** $$X \approx 635,801$$ This is the value of each intermediate payment $X$ to reach the final annuity value of €2,400,000 after 3 periods at 11.5% annual discount rate.