Suppose you invest \( \$ 150 \) a month for 5 years into an account earning \( 8 \% \) compounded monthly. After 5 years, you leave the money, without making additional deposits, in the account for another 30 years. How much will you have in the end?

We first break the problem into two phases. **Phase 1: Depositing \$150 per month for 5 years** Since the deposits are made monthly for 5 years, the number of deposits is \[ n=5\times 12=60. \] The monthly interest rate is \[ i=\frac{8\%}{12}=\frac{0.08}{12}\approx0.0066667. \] The future value of an annuity (regular deposits) is given by \[ FV_{\text{annuity}} = 150 \cdot \frac{\left(1+i\right)^n-1}{i}. \] Thus, after 5 years, \[ FV_{5} = 150 \cdot \frac{(1+0.08/12)^{60}-1}{0.08/12}. \] **Phase 2: Letting the lump sum grow for 30 additional years** After the 5-year period, no additional deposits are made. The amount will continue to earn interest for 30 years (which is \(30\times12=360\) months). The growth factor is therefore \[ (1+i)^{360}. \] So the final account value is \[ FV_{\text{final}} = FV_{5}\cdot (1+i)^{360}. \] Substituting, we obtain \[ FV_{\text{final}} = 150 \cdot \frac{(1+0.08/12)^{60}-1}{0.08/12} \cdot (1+0.08/12)^{360}. \] **Numerical Approximation** 1. Compute the monthly factor: \[ 1+i = 1+\frac{0.08}{12} \approx 1.0066667. \] 2. For 60 months: \[ (1.0066667)^{60}\approx1.4887. \] 3. Then the annuity part: \[ FV_{5}=150\cdot\frac{1.4887-1}{0.0066667} \approx150\cdot\frac{0.4887}{0.0066667}\approx150\cdot73.305\approx10\,995. \] 4. For 360 months: \[ (1.0066667)^{360}\approx10.85. \] 5. Multiplying: \[ FV_{\text{final}} \approx10\,995\times10.85\approx119\,200. \] Thus, the final amount after 35 years is approximately \$119,200.