1. **Problem Statement:** A couple wants to accumulate 100000 by making monthly payments of 1600 into an account earning 9% annual interest compounded monthly. We need to find the number of full payments and the size of the final payment. 2. **Variables in financial calculator terms:** - Future Value (FV) = 100000 - Payment (PMT) = 1600 - Interest rate per period (I/Y) = 9% annual / 12 = 0.75% monthly - Number of periods (N) = ? (to find) - Present Value (PV) = 0 (starting from zero) 3. **Formula for future value of an ordinary annuity:** $$FV = PMT \times \frac{(1 + i)^N - 1}{i}$$ where $i$ is the monthly interest rate and $N$ is the number of payments. 4. **Calculate $N$:** Rearranging the formula to solve for $N$: $$\frac{FV \times i}{PMT} + 1 = (1 + i)^N$$ Taking natural logarithm on both sides: $$N = \frac{\ln\left(\frac{FV \times i}{PMT} + 1\right)}{\ln(1 + i)}$$ 5. **Substitute values:** $$i = \frac{9}{100 \times 12} = 0.0075$$ $$\frac{FV \times i}{PMT} + 1 = \frac{100000 \times 0.0075}{1600} + 1 = 1.46875$$ $$N = \frac{\ln(1.46875)}{\ln(1.0075)} \approx \frac{0.384}{0.00747} \approx 51.4$$ 6. **Interpretation:** The couple needs 51 full payments and a partial final payment (since $N$ is not an integer). 7. **Calculate the amount accumulated after 51 payments:** $$FV_{51} = 1600 \times \frac{(1.0075)^{51} - 1}{0.0075}$$ Calculate $(1.0075)^{51} \approx e^{51 \times 0.00747} = e^{0.38097} \approx 1.4635$ $$FV_{51} = 1600 \times \frac{1.4635 - 1}{0.0075} = 1600 \times \frac{0.4635}{0.0075} = 1600 \times 61.8 = 98880$$ 8. **Remaining amount to reach 100000:** $$100000 - 98880 = 1120$$ 9. **Calculate the final payment:** The final payment is the amount needed to reach 100000 after 51 full payments, so it is 1120. **Final answer:** - Number of full payments: 51 - Final payment: 1120