1. **Problem statement:** Natalie invests 2100 every 6 months for 11 years. For the first 8 years, the interest rate is 3.20% compounded semi-annually. For the next 3 years, the rate is 5.50% compounded semi-annually. 2. **Calculate accumulated value after 8 years:** - Number of periods $n_1 = 8 \times 2 = 16$ (since semi-annual) - Interest rate per period $i_1 = \frac{3.20}{100} \div 2 = 0.016$ - Using the future value of an ordinary annuity formula: $$FV_1 = P \times \frac{(1+i_1)^{n_1} - 1}{i_1}$$ - Substitute values: $$FV_1 = 2100 \times \frac{(1+0.016)^{16} - 1}{0.016}$$ - Calculate: $$ (1.016)^{16} = 1.29356$$ $$FV_1 = 2100 \times \frac{1.29356 - 1}{0.016} = 2100 \times 18.3475 = 38,556.52$$ 3. **Calculate accumulated value after 11 years:** - For the next 3 years, number of periods $n_2 = 3 \times 2 = 6$ - Interest rate per period $i_2 = \frac{5.50}{100} \div 2 = 0.0275$ - The amount $FV_1$ will grow for 6 periods at rate $i_2$: $$FV_1 \times (1+i_2)^{n_2} = 38,556.52 \times (1.0275)^6 = 38,556.52 \times 1.1746 = 45,282.44$$ - Additionally, new payments of 2100 are made for 6 periods at rate $i_2$: $$FV_2 = 2100 \times \frac{(1+i_2)^{n_2} - 1}{i_2} = 2100 \times \frac{1.1746 - 1}{0.0275} = 2100 \times 6.347 = 13,329.54$$ - Total accumulated value after 11 years: $$FV = 45,282.44 + 13,329.54 = 58,611.98$$ 4. **Calculate interest earned:** - Total amount invested: $$2100 \times (11 \times 2) = 2100 \times 22 = 46,200$$ - Interest earned: $$58,611.98 - 46,200 = 12,411.98$$ 5. **Final answers:** - a) Accumulated value after 8 years: $38,556.52$ - b) Accumulated value after 11 years: approximately $58,611.98$ - c) Interest earned: approximately $12,411.98$