Multiple Choice Question Which of the following will result in the highest future value? O These would be the same Making a $$ \$ 1,000 $$ annuity for 10 years starting today at $$ 4 \% $$ and then stopping leaving the amount to grow for 10 more years $$ \$ 1,000 $$ annuity for 10 years starting in 5 years at $$ 4 \% $$ and then stopping leaving the amount to grow for 10 more years

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UpStudy AI · Accepted Answer

Calculate the value by following steps:
- step0: Calculate:
  $$\frac{1000\left(\left(1+0.04\right)^{10}-1\right)}{0.04}$$
- step1: Add the numbers:
  $$\frac{1000\left(1.04^{10}-1\right)}{0.04}$$
- step2: Convert the expressions:
  $$\frac{1000\left(\left(\frac{26}{25}\right)^{10}-1\right)}{0.04}$$
- step3: Subtract the numbers:
  $$\frac{1000\times \frac{26^{10}-25^{10}}{25^{10}}}{0.04}$$
- step4: Multiply the numbers:
  $$\frac{\frac{8\times 26^{10}-8\times 25^{10}}{5^{17}}}{0.04}$$
- step5: Convert the expressions:
  $$\frac{\frac{8\times 26^{10}-8\times 25^{10}}{5^{17}}}{\frac{1}{25}}$$
- step6: Multiply by the reciprocal:
  $$\frac{8\times 26^{10}-8\times 25^{10}}{5^{17}}\times 25$$
- step7: Rewrite the expression:
  $$\frac{8\times 26^{10}-8\times 25^{10}}{5^{17}}\times 5^{2}$$
- step8: Reduce the numbers:
  $$\frac{8\times 26^{10}-8\times 25^{10}}{5^{15}}\times 1$$
- step9: Multiply:
  $$\frac{8\times 26^{10}-8\times 25^{10}}{5^{15}}$$
Calculate or simplify the expression $$ 1000 * ((1 + 0.04)^{10} - 1) / 0.04 * (1 + 0.04)^{10} $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{1000\left(\left(1+0.04\right)^{10}-1\right)}{0.04}\times \left(1+0.04\right)^{10}$$
- step1: Add the numbers:
  $$\frac{1000\left(1.04^{10}-1\right)}{0.04}\times \left(1+0.04\right)^{10}$$
- step2: Convert the expressions:
  $$\frac{1000\left(\left(\frac{26}{25}\right)^{10}-1\right)}{0.04}\times \left(1+0.04\right)^{10}$$
- step3: Subtract the numbers:
  $$\frac{1000\times \frac{26^{10}-25^{10}}{25^{10}}}{0.04}\times \left(1+0.04\right)^{10}$$
- step4: Add the numbers:
  $$\frac{1000\times \frac{26^{10}-25^{10}}{25^{10}}}{0.04}\times 1.04^{10}$$
- step5: Multiply the numbers:
  $$\frac{\frac{8\times 26^{10}-8\times 25^{10}}{5^{17}}}{0.04}\times 1.04^{10}$$
- step6: Divide the numbers:
  $$\frac{8\times 26^{10}-8\times 25^{10}}{5^{15}}\times 1.04^{10}$$
- step7: Convert the expressions:
  $$\frac{8\times 26^{10}-8\times 25^{10}}{5^{15}}\times \left(\frac{26}{25}\right)^{10}$$
- step8: Evaluate the power:
  $$\frac{8\times 26^{10}-8\times 25^{10}}{5^{15}}\times \frac{26^{10}}{25^{10}}$$
- step9: Multiply the fractions:
  $$\frac{\left(8\times 26^{10}-8\times 25^{10}\right)\times 26^{10}}{5^{15}\times 25^{10}}$$
- step10: Multiply:
  $$\frac{8\times 26^{20}-8\times 650^{10}}{5^{35}}$$
Calculate or simplify the expression $$ 12006.107123 * (1 + 0.04)^{5} $$.
Calculate the value by following steps:
- step0: Calculate:
  $$12006.107123\left(1+0.04\right)^{5}$$
- step1: Add the numbers:
  $$12006.107123\times 1.04^{5}$$
- step2: Convert the expressions:
  $$12006.107123\left(\frac{26}{25}\right)^{5}$$
- step3: Convert the expressions:
  $$\frac{12006107123}{1000000}\left(\frac{26}{25}\right)^{5}$$
- step4: Multiply the numbers:
  $$\frac{12006107123\times 13^{5}}{31250\times 25^{5}}$$
Calculate or simplify the expression $$ 12006.107123 * (1 + 0.04)^{10} $$.
Calculate the value by following steps:
- step0: Calculate:
  $$12006.107123\left(1+0.04\right)^{10}$$
- step1: Add the numbers:
  $$12006.107123\times 1.04^{10}$$
- step2: Convert the expressions:
  $$12006.107123\left(\frac{26}{25}\right)^{10}$$
- step3: Convert the expressions:
  $$\frac{12006107123}{1000000}\left(\frac{26}{25}\right)^{10}$$
- step4: Multiply the numbers:
  $$\frac{192097713968\times 13^{10}}{25^{13}}$$
Calculate or simplify the expression $$ 17771.971453 $$.
Numbers by following steps:
- step0: Solution:
  $$17771.971453$$
Calculate or simplify the expression $$ 14607.265078 $$.
Numbers by following steps:
- step0: Solution:
  $$14607.265078$$
### Summary of Calculations

1. **Option 1: Annuity for 10 years starting today at 4%**
   - Future Value of the annuity after 10 years:
     $$
     FV_1 \approx 12006.11
     $$
   - Future Value after an additional 10 years of growth:
     $$
     FV_{total1} \approx 17771.97
     $$

2. **Option 2: Annuity for 10 years starting in 5 years at 4%**
   - Future Value of the annuity after 10 years (starting in 5 years):
     $$
     FV_2 \approx 12006.11
     $$
   - Future Value after 10 years of growth (only 5 years of growth):
     $$
     FV_{total2} \approx 14607.27
     $$

### Conclusion
- **Option 1** results in a future value of approximately **$17771.97**.
- **Option 2** results in a future value of approximately **$14607.27**.

Thus, the option that results in the highest future value is **Option 1: Making a $1,000 annuity for 10 years starting today at 4% and then stopping, leaving the amount to grow for 10 more years.**
Option 1 results in a higher future value.

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