# RE4 | perplexity/sonar-pro | run 1

## Prompt

You deposit $1,000 at 5% interest compounded annually. What is the balance after 3 years? Round to the nearest cent and show your work.

## Response

You use the **compound interest formula** \(A = P(1 + r)^t\) for annual compounding and then evaluate it with the given numbers. The balance after 3 years is **\$1,157.63** (rounded to the nearest cent).

## Step 1: Write down the formula

For interest compounded annually, the compound interest formula is:

\[
A = P(1 + r)^t
\]

where:  

- \(A\) = amount (final balance)  
- \(P\) = principal (initial deposit)  
- \(r\) = annual interest rate (as a decimal)  
- \(t\) = time in years  

This is a special case of the general compound interest formula \(A = P(1 + r/n)^{nt}\) when \(n = 1\).[Compound Interest Formula](https://www.calculatorsoup.com/calculators/financial/compound-interest-calculator.php)[Compound Interest Wiki](https://en.wikipedia.org/wiki/Compound_interest)

## Step 2: Substitute the given values

You are given:

- \(P = 1000\)  
- \(r = 5\% = 0.05\)  
- \(t = 3\) years  
- Annual compounding → \(n = 1\), so formula reduces to \(A = P(1 + r)^t\)

Substitute:

\[
A = 1000(1 + 0.05)^3
\]

\[
A = 1000(1.05)^3
\]

## Step 3: Compute the power

First compute \(1.05^3\):

\[
1.05^2 = 1.05 \times 1.05 = 1.1025
\]

\[
1.05^3 = 1.1025 \times 1.05 = 1.157625
\]

## Step 4: Multiply by the principal

\[
A = 1000 \times 1.157625 = 1157.625
\]

## Step 5: Round to the nearest cent

\[
A \approx \$1{,}157.63
\]

So, the **balance after 3 years** is **\$1,157.63**.

Would you like to see how the balance grows year by year to compare compound vs simple interest?
