Interest and TVM You deposit $$ \$ 1000 $$ in an account that pays $$ 7 \% $$ interest compounded semiannually. After 2 years, the interest rate is increased to $$ 7.44 \% $$ compounded quarterly. What will be the value of the account after a total of 4 years? Click the icon to view some finance formulas. The value of the acoount will be $$ \$ \square $$. (Round to the nearest dollar as needed.)

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We start by computing the value of the account after the first 2 years with a $$7\%$$ annual rate compounded semiannually.

1. The interest rate per period (half-year) is
   $$
   \frac{7\%}{2} = 0.035.
   $$
2. The number of compounding periods in 2 years is
   $$
   2 \text{ years} \times 2 = 4.
   $$
3. Thus, the value after 2 years is
   $$
   V_1 = 1000 \times (1 + 0.035)^4.
   $$

Next, for the subsequent 2 years the rate increases to $$7.44\%$$ compounded quarterly.

1. The interest rate per period (quarter) is
   $$
   \frac{7.44\%}{4} = \frac{0.0744}{4} = 0.0186.
   $$
2. The number of compounding periods in 2 years is
   $$
   2 \text{ years} \times 4 = 8.
   $$
3. The final value of the account after the next 2 years is
   $$
   V_2 = V_1 \times (1 + 0.0186)^8.
   $$

Substitute $$V_1$$ into $$V_2$$:
$$
V_2 = 1000 \times (1.035)^4 \times (1.0186)^8.
$$

Now, we compute the values step by step:

1. Calculate $$ (1.035)^4 $$:
   $$
   (1.035)^4 \approx 1.1477.
   $$
2. Calculate $$ (1.0186)^8 $$:
   $$
   (1.0186)^8 \approx 1.1587.
   $$

Then,
$$
V_2 \approx 1000 \times 1.1477 \times 1.1587 \approx 1000 \times 1.3310 = 1331.0.
$$

Rounding to the nearest dollar, the final value of the account is
$$
\$1331.
$$
The account will be worth \$1,331 after 4 years.

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