Solutions 1. a) $$ \mathrm{A}=\mathrm{P}(1+i)^{n} $$ $$ \mathrm{~A}=? \quad \mathrm{P}=\mathrm{R} 1570 $$

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Certainly! Let's work through the solution step by step based on the information provided.

### Problem Statement:
You have the formula for calculating the future amount $$ \mathrm{A} $$ using compound interest:

$$
\mathrm{A} = \mathrm{P} \times (1 + i)^n
$$

Where:
- $$ \mathrm{A} $$ = the amount of money accumulated after $$ n $$ periods, including interest.
- $$ \mathrm{P} $$ = the principal amount (initial investment).
- $$ i $$ = the interest rate per period.
- $$ n $$ = the number of periods the money is invested for.

Given:
- $$ \mathrm{P} = \mathrm{R}\,1570 $$

### To Find:
$$ \mathrm{A} = ? $$

### Solution:
To determine the future amount $$ \mathrm{A} $$, we need to know two additional pieces of information:
1. **Interest Rate ($$ i $$)**: The percentage at which your investment grows each period.
2. **Number of Periods ($$ n $$)**: How many times the interest is applied.

Since these values aren't provided, I'll outline the steps you would follow once you have them.

#### Step 1: Identify the Values
- **Principal ($$ \mathrm{P} $$)**: R1570
- **Interest Rate ($$ i $$)**: *(e.g., 5% per annum, which is 0.05 in decimal)*
- **Number of Periods ($$ n $$)**: *(e.g., 3 years)*

#### Step 2: Convert Percentage to Decimal (if necessary)
If the interest rate is given as a percentage, convert it to a decimal by dividing by 100.

$$
i = \frac{\text{Interest Rate (\%)}}{100}
$$

*Example:* If $$ i = 5\% $$, then $$ i = 0.05 $$.

#### Step 3: Plug the Values into the Formula
$$
\mathrm{A} = \mathrm{P} \times (1 + i)^n
$$

*Using the example values:*
$$
\mathrm{A} = 1570 \times (1 + 0.05)^3
$$

#### Step 4: Calculate the Amount

1. Calculate $$ (1 + i)^n $$:
   $$
   (1 + 0.05)^3 = 1.157625
   $$
2. Multiply by the principal:
   $$
   \mathrm{A} = 1570 \times 1.157625 \approx 1812.31
   $$

*So, after 3 years at an annual interest rate of 5%, the amount $$ \mathrm{A} $$ would be approximately R1812.31.*

### Final Answer:
$$
\mathrm{A} = \mathrm{P} \times (1 + i)^n = 1570 \times (1 + i)^n
$$

*To find the numerical value of $$ \mathrm{A} $$, substitute the specific values for the interest rate $$ i $$ and the number of periods $$ n $$ into the equation.*

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**Note:** If you provide the interest rate and the number of periods, I can help you calculate the exact value of $$ \mathrm{A} $$.
To find $$ \mathrm{A} $$, use the formula $$ \mathrm{A} = \mathrm{P} \times (1 + i)^n $$. Plug in the principal $$ \mathrm{P} = \mathrm{R}\,1570 $$ and the known interest rate and number of periods to calculate $$ \mathrm{A} $$.

Solutions 1. a) \( \mathrm{A}=\mathrm{P}(1+i)^{n} \) \( \mathrm{~A}=? \quad \mathrm{P}=\mathrm{R} 1570 \)

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