1. Strong Induction:
   • Strong induction is a variation of mathematical induction that strengthens the inductive step by assuming that the statement holds true for all natural numbers up to a certain value, rather than just the previous number.
   • In strong induction, to prove that a statement
     
     $$

     P(n) holds true for all natural numbers
     $\mathrm{<span\; class="math\; math-inline"\; style="font-family:\; Georgia,\; \text{'}Times\; New\; Roman\text{'},\; \text{'}Bitstream\; Charter\text{'},\; Times,\; serif;"><span\; class="katex"><span\; class="katex-html"\; aria-hidden="true"><span\; class="base"><span\; class="mord\; mathnormal">n</span><span\; class="mspace"></span><span\; class="mrel">\ge </span><span\; class="mspace"></span></span><span\; class="base"><span\; class="strut"></span><span\; class="mord\; mathnormal">k\; </span></span></span></span>for\; some\; fixed </span>}$

     
     $\mathrm{<br>}$

     k, you need to prove: a. Base Case: Prove that P(k) is true. b. Inductive Step: Assume that 

     P(i) is true for all k≤i≤n, for some n≥k, and then prove that 

     P(n+1) is also true.

   • Strong induction allows for more flexibility in the inductive step, as it assumes the truth of the statement for a larger range of values, potentially simplifying the proof.
   • Strong induction is often used when the statement depends on multiple preceding values or when the inductive hypothesis requires more than just the previous case.
2. Induction with Nonzero Base Cases:
   • In some cases, the base case for mathematical induction may not start at
     
     $$

     n=0 or
     $$

     n=1, but instead, it starts at a nonzero value, say
     $\mathrm{<br>}$

     n=m, where m is some fixed natural number.

   • The process of induction remains the same, but the base case is shifted to
     
     $$

     n=m, and then the inductive step proceeds as usual.

   • For example, to prove a statement
     

     $P(n)$ for all natural numbers 

     
     $$

     $n\ge m$, you would: a. Base Case: Prove that
     $\mathrm{<span\; class="math\; math-inline"\; style="font-family:\; Georgia,\; \text{'}Times\; New\; Roman\text{'},\; \text{'}Bitstream\; Charter\text{'},\; Times,\; serif;"><span\; class="katex"><span\; class="katex-html"\; aria-hidden="true"><span\; class="base"><span\; class="mord\; mathnormal">P</span><span\; class="mopen">(</span><span\; class="mord\; mathnormal">m</span><span\; class="mclose">)</span></span></span></span>is\; true.\; b.</span><strong\; style="font-family:\; Georgia,\; \text{'}Times\; New\; Roman\text{'},\; \text{'}Bitstream\; Charter\text{'},\; Times,\; serif;">Inductive\; Step</strong><span\; style="font-family:\; Georgia,\; \text{'}Times\; New\; Roman\text{'},\; \text{'}Bitstream\; Charter\text{'},\; Times,\; serif;">:\; Assume\; that </span>}$

     
     $\mathrm{<br>}$

     $P(i)$ is true for all $m\le i\le n$and then prove that $P(n+1)$ is also true.

Both strong induction and induction with nonzero base cases provide powerful tools for proving statements about natural numbers and other mathematical structures. They offer flexibility in handling various types of problems and are widely used across different areas of mathematics.