Robbins Rodriquez
11/02/2024 · Primary School

Review of Principle of Mathematical Induction and Pigeonhole Principle 11. Prove that for all natural numbers \( n \), the following holds: \( 1+2+3+\ldots+\mathrm{n}=\left[\frac{n(n+1)}{2}\right] \) 12. Prove that for all \( n \geq 1,2^{n}>n \) 13. In a group of 13 people, show that at least two people must share the same birthday month.

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11. Proven using mathematical induction that the sum of the first \( n \) natural numbers is \( \frac{n(n+1)}{2} \). 12. Proven using mathematical induction that \( 2^n > n \) for all \( n \geq 1 \). 13. Proven using the Pigeonhole Principle that in a group of 13 people, at least two share the same birthday month.

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