Five years

Nakato Nobuki, a Japanese artist, wants to have $24,000.00 in his savings account at the end of five years. Mr. Nobuki deposits $1,500.00 annually into savings and has a balance of $8,000.00 today. What must the interest rate on the savings account be for Nobuki to achieve his goals?

Correct answer:

i =  12.451 %

Step-by-step explanation:

$$\begin{gathered} s=24000 \\ a=1500 \\ b=8000 \\ t=a\cdot 5+b=1500\cdot 5+8000=15500 \\ t<s \\ {r}^5\cdot b+r^4\cdot a+r^4\cdot a+r^3\cdot a+r^2\cdot a+r\cdot a=s \\ r=1.12451\doteq 1.1245 \\ i=100\cdot (r-1)=100\cdot (1.1245-1)=12.451=12.451\% \\ \text{ Verifying Solution: } \\ {x}_1=b\cdot r+a=8000\cdot 1.1245+1500=10496.08 \\ {x}_2=x_1\cdot r+a=10496.08\cdot 1.1245+1500\doteq 13302.9469 \\ {x}_3=x_2\cdot r+a=13302.9469\cdot 1.1245+1500\doteq 16459.2968 \\ {x}_4=x_3\cdot r+a=16459.2968\cdot 1.1245+1500\doteq 20008.6439 \\ {x}_5=x_4\cdot r+a=20008.6439\cdot 1.1245+1500\doteq 23999.9201 \end{gathered}$$

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