Usually, mathematical induction inequality proof requires one initial value, but in some cases, **two initials** are to be required, such as Fibonacci sequence. In this case, it is required to show **two initials** are working as the first step of the mathematical induction inequality proof, and **two assumptions** are to be placed for the third steps.

The following worked example shows how to handle the mathematical induction inequality proof with two initials.

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### Worked Example for Mathematical Induction Inequality Proof with Two Initials

A sequence \(S_n\) is defined by \(S_1 = 1, S_2 = 2\) and for \(n \gt 2, S_n = S_{n-1} + (n – 1) S_{n-2}\).

(a) Prove \(\sqrt{x} + x \ge \sqrt{x(1 + x)} \) for all real numbers \(x \ge 0 \).

(b) Prove \(S_n \ge \sqrt{n!} \) for all integers \(n \ge 1\).

### Related Topics

**Best Examples of Mathematical Induction Inequality**

**Best Examples of Mathematical Induction Divisibility**

**Mathematical Induction Fundamentals**

**Mathematical Induction Inequality Proof with Factorials**