Mathwords logoMathwords

Bernoulli Inequality — Definition, Formula & Examples

Bernoulli Inequality states that for any real number x≥−1x \ge -1 and any positive integer nn, the expression (1+x)n(1 + x)^n is always at least as large as 1+nx1 + nx.

For all x∈Rx \in \mathbb{R} with x≥−1x \ge -1 and for every positive integer n≥1n \ge 1, the inequality (1+x)n≥1+nx(1 + x)^n \ge 1 + nx holds, with equality if and only if n=1n = 1 or x=0x = 0.

Key Formula

(1+x)n≥1+nx(1 + x)^n \ge 1 + nx
Where:
  • xx = A real number satisfying $x \ge -1$
  • nn = A positive integer ($n \ge 1$)

How It Works

The inequality gives a simple linear lower bound for an exponential-type expression. You can prove it by mathematical induction: the base case n=1n = 1 is trivially true, and the inductive step uses the fact that 1+x≥01 + x \ge 0 to preserve the inequality when multiplying both sides. In practice, you apply it whenever you need to estimate (1+x)n(1 + x)^n from below without expanding the full binomial. It is especially powerful when xx is small, since the linear approximation 1+nx1 + nx is then quite close to the true value.

Worked Example

Problem: Use the Bernoulli Inequality to find a lower bound for (1.03)10(1.03)^{10}.
Identify x and n: Write (1.03)10(1.03)^{10} in the form (1+x)n(1 + x)^n with x=0.03x = 0.03 and n=10n = 10. Since 0.03≥−10.03 \ge -1, the inequality applies.
(1+0.03)10≥1+10(0.03)(1 + 0.03)^{10} \ge 1 + 10(0.03)
Compute the lower bound: Evaluate the right-hand side.
1+10(0.03)=1+0.3=1.31 + 10(0.03) = 1 + 0.3 = 1.3
Compare: The actual value of (1.03)10≈1.3439(1.03)^{10} \approx 1.3439, so the bound 1.31.3 is correct and reasonably tight.
Answer: (1.03)10≥1.3(1.03)^{10} \ge 1.3

Why It Matters

Bernoulli Inequality appears frequently in mathematical olympiads for bounding products and powers. It also serves as a stepping stone toward proving the AM-GM inequality and understanding convergence results in calculus and real analysis.

Common Mistakes

Mistake: Applying the inequality when x<−1x < -1, for example using x=−2x = -2.
Correction: The condition x≥−1x \ge -1 is essential. When x<−1x < -1, the factor (1+x)(1 + x) is negative and raising it to various powers can flip the inequality direction. Always verify x≥−1x \ge -1 before applying.