1. Problem: Sara wants to know how long it will take to have a total of 175000 kr. on her account if she continues to deposit 3000 kr. monthly into an account with a monthly interest rate of 0.4% after moving her initial 50000 kr. to this account. 2. Formula: For monthly deposits with compound interest, the future value $FV$ after $n$ months is given by: $$FV = P \times (1 + r)^n + PMT \times \frac{(1 + r)^n - 1}{r}$$ where: - $P$ is the initial principal (50000 kr.) - $r$ is the monthly interest rate (0.4% = 0.004) - $PMT$ is the monthly deposit (3000 kr.) - $n$ is the number of months 3. We want to find $n$ such that: $$175000 = 50000 \times (1 + 0.004)^n + 3000 \times \frac{(1 + 0.004)^n - 1}{0.004}$$ 4. Let $x = (1 + 0.004)^n$. Then: $$175000 = 50000x + 3000 \times \frac{x - 1}{0.004} = 50000x + 750000(x - 1) = 50000x + 750000x - 750000 = 800000x - 750000$$ 5. Solve for $x$: $$175000 + 750000 = 800000x$$ $$925000 = 800000x$$ $$x = \frac{925000}{800000} = 1.15625$$ 6. Recall $x = (1.004)^n$, so: $$1.004^n = 1.15625$$ Take natural logarithm on both sides: $$n \ln(1.004) = \ln(1.15625)$$ $$n = \frac{\ln(1.15625)}{\ln(1.004)}$$ 7. Calculate values: $$\ln(1.15625) \approx 0.1451$$ $$\ln(1.004) \approx 0.003992$$ 8. Finally: $$n = \frac{0.1451}{0.003992} \approx 36.34$$ 9. Interpretation: It takes about 36.34 months, or approximately 3 years and 1 month, for Sara to reach 175000 kr. on the account. Answer: Sara will have 175000 kr. after about 36 months of monthly deposits and interest.