Proof by mathematical induction

• “Proof by mathematical induction” is a method to establish the validity of a given statement for all natural numbers.
• This method involves two steps: Base Case, and Inductive Step.
• In the “base case”, we test the statement for the smallest value, usually when n = 1.
• The “inductive step” consists of two parts: assumption (inductive hypothesis), and show it to be true for n+1.
• In the assumption part, we assume the statement is true for some arbitrary natural number n.
• In the next part, we need to show that, given the statement holds for a particular n, it also holds for n+1.
• If both the base case and the inductive step are proven, then by induction, the statement is true for all natural numbers.
• Keep in mind the concept of “domino effect” in understanding the principle of mathematical induction. If the first domino (base case) falls, and each domino knocks over the next (inductive step), all dominos will fall.
• Mathematical induction finds broad application in number theory and proof of formulas.
• In Core Pure Mathematics 1, you will have to apply the principles of Mathematical Induction to prove series sum formulae, divisibility, inequalities, etc.
• While performing a mathematical induction proof, ensure to: Write the statement for n and n+1 clearly, prove that the statement is true for n = 1 (or some other suitable beginning point), and then assume that the statement is true for any value of n and prove it for n+1.
• Be patient with induction problems. They often require a mix of algebraic manipulation and creativity.