Where \( P = 1000 \), \( r = 0.05 \), and \( n = 3 \). - Coaching Toolbox

Understanding the Context

Compound interest is the interest calculated on the initial principal and also on the accumulated interest from previous periods. Unlike simple interest—where interest is earned only on the principal—compound interest accelerates growth exponentially.

The Formula for Compound Interest

The formula to compute the future value ( A ) of an investment is:

Key Insights

[
A = P \left(1 + rac{r}{n} ight)^{nt}
]

Where:
- ( A ) = the amount of money accumulated after ( t ) years, including interest
- ( P ) = principal amount ($1000)
- ( r ) = annual interest rate (5% = 0.05)
- ( n ) = number of times interest is compounded per year (3)
- ( t ) = number of years the money is invested (in this example, we’ll solve for a variable time)

Solving for Different Time Periods

Since ( t ) isn’t fixed, let’s see how ( A ) changes over 1, 3, and 5 years under these settings.

Final Thoughts

Case 1: ( t = 1 ) year
[
A = 1000 \left(1 + rac{0.05}{3}

ight)^{3 \ imes 1} = 1000 \left(1 + 0.016667 ight)^3 = 1000 \ imes (1.016667)^3 pprox 1000 \ imes 1.050938 = $1050.94
]

Case 2: ( t = 3 ) years
[
A = 1000 \left(1 + rac{0.05}{3}

ight)^{9} = 1000 \ imes (1.016667)^9 pprox 1000 \ imes 1.161472 = $1161.47
]

Case 3: ( t = 5 ) years
[
A = 1000 \left(1 + rac{0.05}{3}

ight)^{15} = 1000 \ imes (1.016667)^{15} pprox 1000 \ imes 1.283357 = $1283.36
]

Key Takeaways

• At ( P = 1000 ) and ( r = 5% ), compounding ( n = 3 ) times per year leads to steady growth, with the amount almost 28.6% higher after 5 years compared to a single compounding cycle.
  - Because interest is applied multiple times per year, even a modest rate compounds significantly over time.
  - This timing formula helps users plan savings, loans, and investments by projecting returns with different compounding frequencies and durations.