State and prove the principles of mathematical induction.

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6) State the principles of mathematical induction
Mathematical induction is a proof technique used to establish that a statement $P(n)$ is true for all natural numbers $n$. It consists of two main steps:

• Base Case: Show that the statement $P(n)$ is true for the initial value of $n$ (usually $n=1$).
• Inductive Step: Assume that the statement $P(k)$ is true for an arbitrary natural number $k \ge 1$ (this is the inductive hypothesis). Then, prove that $P(k+1)$ is also true.

**7) Prove by mathematical induction that;