1. **State the problem:** Amir deposits 10,000 at the beginning of each year for 13 years in an account paying 8% compounded annually. Then he transfers the total amount to another account paying 9% compounded semiannually for 8 more years. We need to find the final amount after 21 years. 2. **First phase: Annuity due calculation** Since deposits are at the beginning of each year, this is an annuity due. The future value of an annuity due is given by: $$FV = P \times \frac{(1 + r)^n - 1}{r} \times (1 + r)$$ where $P=10000$, $r=0.08$, $n=13$. 3. **Calculate the future value after 13 years:** $$FV = 10000 \times \frac{(1 + 0.08)^{13} - 1}{0.08} \times (1 + 0.08)$$ Calculate $(1 + 0.08)^{13}$: $$1.08^{13} \approx 2.718186$$ Then: $$\frac{2.718186 - 1}{0.08} = \frac{1.718186}{0.08} = 21.477325$$ Multiply by $10000$ and then by $1.08$: $$10000 \times 21.477325 \times 1.08 = 10000 \times 23.195291 = 231952.91$$ So, after 13 years, the amount is approximately $231,952.91$. 4. **Second phase: Compound interest for 8 years at 9% compounded semiannually** The formula for compound interest is: $$A = P \times \left(1 + \frac{r}{m}\right)^{mt}$$ where $P=231952.91$, $r=0.09$, $m=2$ (semiannual), $t=8$. Calculate: $$A = 231952.91 \times \left(1 + \frac{0.09}{2}\right)^{2 \times 8} = 231952.91 \times (1.045)^{16}$$ Calculate $(1.045)^{16}$: $$1.045^{16} \approx 2.025825$$ Multiply: $$231952.91 \times 2.025825 = 469,657.88$$ 5. **Final answer:** The final amount on deposit after the entire 21-year period is approximately **469,657.88**.