Deposit for house

The current house price is $300000, with prices increasing by 3% yearly. A buyer wishes to purchase the house in 5 years. The bank requires a 10%deposit on the price of the house in order to grant a loan. How much would the buyer need to deposit each quarter in an ordinary annuity earning 8% interest per annum compounded quarterly to be able to have the amount required for the deposit in 5 years?

Correct answer:

x =  1431.36 USD

Step-by-step explanation:

$$\begin{gathered} a=300000\text{USD} \\ q=100\%+3\%=1+\frac{3}{100}=1.03 \\ b=a\cdot q^5=300000\cdot 1.03^5\doteq 347782.2223\text{USD} \\ {q}_1=10\%=\frac{10}{100}=\frac{1}{10}=0.1 \\ d=q_1\cdot b=0.1\cdot 347782.2223\doteq 34778.2222\text{USD} \\ p=8\%=\frac{8}{100}=\frac{2}{25}=0.08 \\ r=1+p/4=1+0.08/4=\frac{51}{50}=1.02 \\ {r}_1=1 \\ {r}_2=r_1\cdot r+1=1\cdot 1.02+1=\frac{101}{50}=2.02 \\ {r}_3=r_2\cdot r+1=2.02\cdot 1.02+1=3.0604 \\ {r}_4=r_3\cdot r+1=3.0604\cdot 1.02+1\doteq 4.1216 \\ {r}_5=r_4\cdot r+1=4.1216\cdot 1.02+1\doteq 5.204 \\ {r}_6=r_5\cdot r+1=5.204\cdot 1.02+1\doteq 6.3081 \\ {r}_7=r_6\cdot r+1=6.3081\cdot 1.02+1\doteq 7.4343 \\ {r}_8=r_7\cdot r+1=7.4343\cdot 1.02+1\doteq 8.583 \\ {r}_9=r_8\cdot r+1=8.583\cdot 1.02+1\doteq 9.7546 \\ {r}_{10}=r_9\cdot r+1=9.7546\cdot 1.02+1\doteq 10.9497 \\ {r}_{11}=r_{10}\cdot r+1=10.9497\cdot 1.02+1\doteq 12.1687 \\ {r}_{12}=r_{11}\cdot r+1=12.1687\cdot 1.02+1\doteq 13.4121 \\ {r}_{13}=r_{12}\cdot r+1=13.4121\cdot 1.02+1\doteq 14.6803 \\ {r}_{14}=r_{13}\cdot r+1=14.6803\cdot 1.02+1\doteq 15.9739 \\ {r}_{15}=r_{14}\cdot r+1=15.9739\cdot 1.02+1\doteq 17.2934 \\ {r}_{16}=r_{15}\cdot r+1=17.2934\cdot 1.02+1\doteq 18.6393 \\ {r}_{17}=r_{16}\cdot r+1=18.6393\cdot 1.02+1\doteq 20.0121 \\ {r}_{18}=r_{17}\cdot r+1=20.0121\cdot 1.02+1\doteq 21.4123 \\ {r}_{19}=r_{18}\cdot r+1=21.4123\cdot 1.02+1\doteq 22.8406 \\ {r}_{20}=r_{19}\cdot r+1=22.8406\cdot 1.02+1\doteq 24.2974 \\ x\cdot r_{20}=d \\ x=d/r_{20}=34778.2222/24.2974=1431.36\text{USD} \end{gathered}$$

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