Calculate 5770

Calculate how long it will take the Šetrný family to repay the loan if, after the first year, they paid €1350 in interest and the loan's interest rate was 6% p.a. They will repay the loan to the bank for €3,000 each year, and the bank will only add interest to the remaining amount. How much more € will they return to the bank than they borrowed?

Correct answer:

n =  10.2521 y
x =  8256.1644 Eur

Step-by-step explanation:

$$\begin{gathered} p=6\% \\ q=1+p/100=1+6/100=\frac{53}{50}=1.06 \\ 1350=6/100a \\ a=1350\cdot 100/6=22500\text{Eur} \\ {a}_1=a\cdot q-3000=22500\cdot 1.06-3000=20850 \\ {a}_2=a_1\cdot q-3000=20850\cdot 1.06-3000=19101 \\ {a}_3=a_2\cdot q-3000=19101\cdot 1.06-3000=17247.06 \\ {a}_4=a_3\cdot q-3000=17247.06\cdot 1.06-3000=15281.8836 \\ {a}_5=a_4\cdot q-3000=15281.8836\cdot 1.06-3000\doteq 13198.7966 \\ {a}_6=a_5\cdot q-3000=13198.7966\cdot 1.06-3000\doteq 10990.7244 \\ {a}_7=a_6\cdot q-3000=10990.7244\cdot 1.06-3000\doteq 8650.1679 \\ {a}_8=a_7\cdot q-3000=8650.1679\cdot 1.06-3000\doteq 6169.178 \\ {a}_9=a_8\cdot q-3000=6169.178\cdot 1.06-3000\doteq 3539.3286 \\ {a}_{10}=a_9\cdot q-3000=3539.3286\cdot 1.06-3000\doteq 751.6883 \\ d=\lceil a_{10}/3000\cdot 365\rceil =\lceil 751.6883/3000\cdot 365\rceil =92 \\ n=10+d/365=10+92/365=10.2521\text{y} \end{gathered}$$

$x=n\cdot 3000-a=10.2521\cdot 3000-22500=8256.1644\text{Eur}$

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