Retirement saving

Deryl wishes to save money to provide for his retirement. Beginning one year from now, he will begin depositing the same fixed amount each year for the next 30 years into a retirement savings account. Starting one year after making his last deposit, he will withdraw $100,000 annually for the following 25 years (i.e., he will make 25 withdrawals in all). Assume that the retirement fund earns 12% annually over the period he is depositing money and the period he makes withdrawals. For Deryl to have sufficient funds in his account to fund his retirement, how much should he deposit annually (rounded to the nearest dollar)?

Correct answer:

x =  3640 USD

Step-by-step explanation:

$$\begin{gathered} q=100\%+12\%=1+\frac{12}{100}=1.12 \\ n=30 \\ m=25 \\ w=100000\text{USD/a} \\ {k}_1=\frac{q^n-1}{q-1}=\frac{{\frac{28}{25}}^{30}-1}{\frac{28}{25}-1}\doteq 241.3327 \\ {k}_2=\frac{q^m-1}{q-1}=\frac{{\frac{28}{25}}^{25}-1}{\frac{28}{25}-1}\doteq 133.3339 \\ {k}_3=q^{m-1}=1.12^{25-1}\doteq 15.1786 \\ x\cdot k_1\cdot q^{m-1}=w\cdot k_2 \\ x=\frac{k_2}{k_1\cdot k_3}\cdot w=\frac{133.3339}{241.3327\cdot 15.1786}\cdot 100000\doteq 3639.9197\text{USD} \\ \text{ Verifying Solution: } \\ {s}_1=x\cdot k_1=3639.9197\cdot 241.3327\doteq 878431.5806\text{USD} \\ (((((((((((((((((((((((((X\cdot k_1-w)\cdot q-w)\cdot q-w)\cdot q-w)\cdot q-w)\cdot q-w)\cdot q-w)\cdot q-w)\cdot q-w)\cdot q-w)\cdot q-w)\cdot q-w)\cdot q-w)\cdot q-w)\cdot q-w)\cdot q-w)\cdot q-w)\cdot q-w)\cdot q-w)\cdot q-w)\cdot q-w)\cdot q-w)\cdot q-w)\cdot q-w)\cdot q-w)\cdot q=0 \\ (((((((((((((((((((((((((X\cdot 241.33268434093-100000)\cdot 1.12-100000)\cdot 1.12-100000)\cdot 1.12-100000)\cdot 1.12-100000)\cdot 1.12-100000)\cdot 1.12-100000)\cdot 1.12-100000)\cdot 1.12-100000)\cdot 1.12-100000)\cdot 1.12-100000)\cdot 1.12-100000)\cdot 1.12-100000)\cdot 1.12-100000)\cdot 1.12-100000)\cdot 1.12-100000)\cdot 1.12-100000)\cdot 1.12-100000)\cdot 1.12-100000)\cdot 1.12-100000)\cdot 1.12-100000)\cdot 1.12-100000)\cdot 1.12-100000)\cdot 1.12-100000)\cdot 1.12-100000)\cdot 1.12=0 \\ 4102.671177X=14933393.4462 \\ X=\frac{14933393.4462}{4102.67117723}=3639.91965262 \\ X=3639.919653 \end{gathered}$$

Try another example