\( 1< \) Decide whether the following statement makes sense (or is clearly true) or does not make sense (or is clearly false), Explain your reasoning. If interest rates stay at \( 5 \% \) APR and I continue to make my monithly \( \$ 25 \) deposits into my retirement plan, I should have at least \( \$ 30,000 \) saved viten I retire in 35 years. The statement (Round to the nearest cent as needed.)

We are given that a person makes monthly deposits of \( \$25 \) for 35 years with an interest rate of \( 5\% \) APR. We wish to know whether the accumulated amount will be at least \( \$30,\!000 \) at retirement. **Step 1. Determine the number of deposits** Since deposits are monthly for 35 years, the total number of deposits is \[ n = 35 \times 12 = 420. \] **Step 2. Determine the monthly interest rate** The APR is \( 5\% \); thus the monthly interest rate is \[ r = \frac{5\%}{12} = \frac{0.05}{12} \approx 0.0041667. \] **Step 3. Write the future value formula for an annuity** The future value \( FV \) of an annuity with a payment \( P \) each period, interest rate \( r \) per period, and \( n \) periods is given by \[ FV = P \times \frac{(1 + r)^n - 1}{r}. \] Here, \( P = 25 \), \( r \approx 0.0041667 \), and \( n = 420 \). **Step 4. Substitute the values into the formula** \[ FV = 25 \times \frac{(1 + 0.0041667)^{420} - 1}{0.0041667}. \] **Step 5. Compute the accumulation factor** We first calculate \((1 + r)^n\): \[ (1.0041667)^{420} \approx e^{420 \ln(1.0041667)}. \] Calculating the natural logarithm: \[ \ln(1.0041667) \approx 0.0041589, \] so \[ 420 \times 0.0041589 \approx 1.7487. \] Then \[ (1.0041667)^{420} \approx e^{1.7487} \approx 5.75. \] **Step 6. Complete the future value calculation** Now, substitute this back into the formula: \[ FV \approx 25 \times \frac{5.75 - 1}{0.0041667} = 25 \times \frac{4.75}{0.0041667}. \] Next, compute the division: \[ \frac{4.75}{0.0041667} \approx 1140. \] Thus, \[ FV \approx 25 \times 1140 \approx 28,\!500. \] **Step 7. Compare with the claimed amount** The computed future value is approximately \( \$28,\!500 \), which is **less than** the claimed \( \$30,\!000 \). **Conclusion** The statement "If interest rates stay at \( 5\% \) APR and I continue to make my monthly \( \$25 \) deposits into my retirement plan, I should have at least \( \$30,\!000 \) saved when I retire in 35 years" is clearly false because the calculations show an accumulated value of approximately \( \$28,\!500 \), which is below \( \$30,\!000 \).