Mathematical Induction Inequality Proof with Two Initials

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 \(n \ge 1\).

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