Find the future value for the ordinary annuity with the given payment and interest rate. PMT $$ =\$ 2,250 ; 1.40 \% $$ compounded quarterly for 8 years. The future value of the ordinary annuity is $$ \$ \square $$. (Do not round until the final answer. Then round to the nearest cent as needed.)

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To find the future value of an ordinary annuity, we can use the formula:

$$
FV = PMT \times \frac{(1 + r/n)^{nt} - 1}{r/n}
$$

Where:
- $$ FV $$ = future value of the annuity
- $$ PMT $$ = payment amount per period
- $$ r $$ = annual interest rate (as a decimal)
- $$ n $$ = number of compounding periods per year
- $$ t $$ = number of years

Given:
- $$ PMT = 2250 $$
- $$ r = 1.40\% = 0.014 $$
- $$ n = 4 $$ (since it is compounded quarterly)
- $$ t = 8 $$

Now, we can substitute these values into the formula step by step.

1. Calculate $$ r/n $$:
   $$
   r/n = \frac{0.014}{4}
   $$

2. Calculate $$ nt $$:
   $$
   nt = 4 \times 8
   $$

3. Calculate $$ (1 + r/n)^{nt} $$:
   $$
   (1 + r/n)^{nt} = \left(1 + \frac{0.014}{4}\right)^{32}
   $$

4. Finally, substitute everything back into the future value formula.

Let's perform these calculations step by step.
Calculate the value by following steps:
- step0: Calculate:
  $$4\times 8$$
- step1: Multiply the numbers:
  $$32$$
Calculate or simplify the expression $$ (1 + 0.014/4)^(4*8) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\left(1+\frac{0.014}{4}\right)^{4\times 8}$$
- step1: Divide the terms:
  $$\left(1+\frac{7}{2000}\right)^{4\times 8}$$
- step2: Add the numbers:
  $$\left(\frac{2007}{2000}\right)^{4\times 8}$$
- step3: Multiply the numbers:
  $$\left(\frac{2007}{2000}\right)^{32}$$
- step4: Use the properties of exponents:
  $$\frac{2007^{32}}{2000^{32}}$$
Calculate or simplify the expression $$ 0.014/4 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{0.014}{4}$$
- step1: Convert the expressions:
  $$\frac{\frac{7}{500}}{4}$$
- step2: Multiply by the reciprocal:
  $$\frac{7}{500}\times \frac{1}{4}$$
- step3: Multiply the fractions:
  $$\frac{7}{500\times 4}$$
- step4: Multiply:
  $$\frac{7}{2000}$$
Calculate or simplify the expression $$ 2250 * ((1 + 0.0035)^(32) - 1) / 0.0035 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{2250\left(\left(1+0.0035\right)^{32}-1\right)}{0.0035}$$
- step1: Add the numbers:
  $$\frac{2250\left(1.0035^{32}-1\right)}{0.0035}$$
- step2: Convert the expressions:
  $$\frac{2250\left(\left(\frac{2007}{2000}\right)^{32}-1\right)}{0.0035}$$
- step3: Subtract the numbers:
  $$\frac{2250\times \frac{2007^{32}-2000^{32}}{2000^{32}}}{0.0035}$$
- step4: Multiply the numbers:
  $$\frac{\frac{9\times 2007^{32}-9\times 2000^{32}}{250^{31}\times 8^{32}}}{0.0035}$$
- step5: Convert the expressions:
  $$\frac{\frac{9\times 2007^{32}-9\times 2000^{32}}{250^{31}\times 8^{32}}}{\frac{7}{2000}}$$
- step6: Multiply by the reciprocal:
  $$\frac{9\times 2007^{32}-9\times 2000^{32}}{250^{31}\times 8^{32}}\times \frac{2000}{7}$$
- step7: Rewrite the expression:
  $$\frac{9\times 2007^{32}-9\times 2000^{32}}{250^{31}\times 8^{32}}\times \frac{250\times 8}{7}$$
- step8: Reduce the numbers:
  $$\frac{9\times 2007^{32}-9\times 2000^{32}}{250^{30}\times 8^{32}}\times \frac{8}{7}$$
- step9: Rewrite the expression:
  $$\frac{9\times 2007^{32}-9\times 2000^{32}}{2^{30}\times 125^{30}\times 8^{32}}\times \frac{8}{7}$$
- step10: Rewrite the expression:
  $$\frac{9\times 2007^{32}-9\times 2000^{32}}{2^{30}\times 125^{30}\times 8^{32}}\times \frac{2^{3}}{7}$$
- step11: Reduce the numbers:
  $$\frac{9\times 2007^{32}-9\times 2000^{32}}{2^{27}\times 125^{30}\times 8^{32}}\times \frac{1}{7}$$
- step12: Multiply the fractions:
  $$\frac{9\times 2007^{32}-9\times 2000^{32}}{2^{123}\times 125^{30}\times 7}$$
- step13: Multiply:
  $$\frac{9\times 2007^{32}-9\times 2000^{32}}{7\times 2^{123}\times 125^{30}}$$
Now that we have completed the calculations, let's summarize the results:

1. The value of $$ r/n $$ is approximately $$ 0.0035 $$.
2. The value of $$ nt $$ is $$ 32 $$.
3. The value of $$ (1 + r/n)^{nt} $$ is approximately $$ 1.118294 $$.

Now, we can substitute these values into the future value formula:

$$
FV = 2250 \times \frac{(1.118294 - 1)}{0.0035}
$$

Calculating this gives us:

$$
FV \approx 2250 \times \frac{0.118294}{0.0035} \approx 2250 \times 33.57085714 \approx 76046.248094
$$

Finally, rounding to the nearest cent, the future value of the ordinary annuity is:

$$
FV \approx \$ 76,046.25
$$

Thus, the future value of the ordinary annuity is $$ \$ 76,046.25 $$.
The future value of the ordinary annuity is \$76,046.25.

Find the future value for the ordinary annuity with the given payment and interest rate. PMT \( =\$ 2,250 ; 1.40 \% \) compounded quarterly for 8 years. The future value of the ordinary annuity is \( \$ \square \). (Do not round until the final answer. Then round to the nearest cent as needed.)

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