To calculate the compound interest earned, we need to use the given formula:

$A = P \left( 1 + \frac{r}{m} \right)^{mt}$

Where:

• $P$ = initial investment (in dollars),
• $r$ = yearly interest rate (expressed as a decimal),
• $m$ = number of times the interest is compounded per year,
• $t$ = number of years.

Steps:

1. Substitute the values of $P$, $r$, $m$, and $t$ into the formula to calculate the compound amount $A$.
2. Simplify the expression to get the value of $A$.
3. Subtract the initial investment $P$ from $A$ to get the interest earned.

If you provide specific values for $P$, $r$, $m$, and $t$, I can help you carry out the calculations and find the interest earned. Would you like to proceed with specific values?

Example (without specific values):

Assuming you have:

• $P = 1000$ (investment),
• $r = 0.05$ (5% interest rate),
• $m = 4$ (compounded quarterly),
• $t = 3$ (years),

The formula becomes:

$A = 1000 \left( 1 + \frac{0.05}{4} \right)^{4 \cdot 3} = 1000 \left( 1 + 0.0125 \right)^{12}$

From there, we simplify further to get $A$, and then subtract $P$ to find the interest earned.

Would you like me to continue with this example or with your specific values?

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Here are five follow-up questions:

1. What values of $P$, $r$, $m$, and $t$ are you working with?
2. Do you need help calculating the interest for different compounding periods (e.g., monthly, yearly)?
3. Are you familiar with how to convert interest rates from percentages to decimals?
4. Do you want an explanation of why compound interest grows faster than simple interest?
5. Would you like a graphical representation of how compound interest works over time?

Tip: Remember, the more frequently the interest is compounded, the greater the total compound amount will be!