Compound Interest: A = P(1 + r/n)^(nt)

What is compound interest?

Compound interest is interest calculated on the principal plus the interest already earned. Because today's interest earns interest tomorrow, the balance grows exponentially — not linearly — over time.

The formula

$A = P \left(1 + \dfrac{r}{n}\right)^{nt}$

where

• $P$ = principal (initial deposit),
• $r$ = nominal annual interest rate (as a decimal),
• $n$ = number of compounding periods per year,
• $t$ = number of years,
• $A$ = final amount.

The interest earned is $I = A - P$.

Step-by-step solution

Setup: $P = 1000$, $r = 0.05$, $n = 12$ (monthly), $t = 10$ years.

Step 1 — Rate per period: $\dfrac{r}{n} = \dfrac{0.05}{12} = 0.0041\overline{6}$

Step 2 — Total number of compounding periods: $nt = 12 \cdot 10 = 120$

Step 3 — Apply the formula: $A = 1000 \left(1 + \dfrac{0.05}{12}\right)^{120} = 1000 \cdot (1.00416\overline{6})^{120}$

Step 4 — Compute: $A \approx 1000 \cdot 1.64701 \approx \boxed{1647.01}$

Step 5 — Interest earned: $I = A - P = 1647.01 - 1000 = 647.01$

Comparison with simple interest

Under simple interest, $I = Prt = 1000 \cdot 0.05 \cdot 10 = 500$, so $A = 1500$. Compound interest earns $147 more — a 29% larger gain on the interest — for the same rate and time. The extra is "interest on interest."

How compounding frequency matters

• Annually ($n=1$): $A = 1000 \cdot 1.05^{10} = 1628.89$
• Quarterly ($n=4$): $A = 1000 \cdot 1.0125^{40} = 1643.62$
• Monthly ($n=12$): $A \approx 1647.01$
• Daily ($n=365$): $A \approx 1648.66$
• Continuously ($n \to \infty$, use $Pe^{rt}$): $A = 1000 e^{0.5} \approx 1648.72$

More frequent compounding helps, but with rapidly diminishing returns past monthly.

Common mistakes

• Using the annual rate $r$ in the exponent directly without dividing by $n$. The per-period rate is $r/n$, not $r$.
• Forgetting to multiply $t$ by $n$ in the exponent. Ten years compounded monthly is 120 periods, not 10.
• Adding $r \cdot t$ to 1 and calling that compound interest. That is simple interest. Compound uses an exponent.

Try it in the visualization

Slide the rate, compounding frequency, and time. Watch the compound-interest curve lift above the straight simple-interest line — and see monthly, quarterly, and continuous compounding stack up.