A mortgage of ​$260 comma 000 is amortized over 25 years by monthly payments of $ 1694.53 . What is the nominal annual rate of interest compounded​ semi-annually?

To find the nominal annual rate of interest compounded semi-annually for a mortgage, we can use the following approach:

### Given:
- Principal $ P = 260,000 $ dollars
- Monthly payment $ M = 1,694.53 $ dollars
- Number of years $ n = 25 $ years
- Number of payments per year $ p = 12 $ (monthly payments)
- Total number of payments $ N = 25 \times 12 = 300 $ payments

### Step 1: Understanding the relationship between monthly payments and interest rate
The monthly mortgage payment is given by the formula:

\[
M = P \times \frac{r(1 + r)^N}{(1 + r)^N - 1}
\]

Where:
- $ M $ is the monthly payment
- $ P $ is the principal loan amount
- $ r $ is the monthly interest rate (which we need to find)
- $ N $ is the total number of payments

### Step 2: Solve for the monthly interest rate $ r $
We need to solve for $ r $ in the formula, which requires iterative or numerical methods because the equation is nonlinear.

### Step 3: Convert the monthly interest rate to the nominal annual rate compounded

Discover how to calculate the nominal annual interest rate compounded semi-annually for a $260,000 mortgage amortized over 25 years with monthly payments of $1694.53. This problem involves financial mathematics, interest rates, and amortization formulas.

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