Here are a few challenging proofs to be done using Principles Of Mathematical Induction. I will be editing this note hereafter, by keeping new Induction Challenges, most probably one new challenge per week. Check these out and post your solutions in the comments below 📝 📬 📮 !

Challenge 1 -> Prove that the \( n^{th} \) term in the Fibonacci series is :-

\( u_{n} = \dfrac {1}{\sqrt {5}}\left[ \left (\dfrac {1+\sqrt {5}}{2}\right) ^{n}-\left( \dfrac {1-\sqrt {5}}{2}\right) ^{n}\right] \)

Challenge 2 -> Prove the rule of transitivity in inequalities using only the definitions and induction principles.

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