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Mathematical Induction

Induction proves a statement for all integers from a base by establishing a base case and an implication from $k$ to $k+1$ . The inductive hypothesis may be used only after the base case and only to prove the next case. The distinction between a formal hypothesis and an intuitive picture is made explicit so that calculations can be justified, not merely patterned.

Worked reasoning

The base starts the logical chain.
The inductive step propagates truth from each case to the next.

What must an induction proof contain?

A base case and an inductive step

Only several examples

Only the case $n=1$

A graph

Which statement best captures the central mathematical idea in Mathematical Induction?

The inductive hypothesis may be used only after the base case and only to prove the next case.

A familiar formula remains valid after its hypotheses, domain or convergence conditions are removed.

A memorised formula is valid even when its conditions are not met.

The diagram or notation replaces the need to justify each mathematical step.

When starting a problem about Mathematical Induction, which move is most reliable?

Write the governing definition and list its hypotheses before calculating in mathematical induction.

Round every value before identifying the mathematical structure.

Choose the longest available formula and substitute immediately.

Assume the desired conclusion and work backwards without checking equivalence.

Which statement is a misconception that must be rejected when working with Mathematical Induction?

A familiar formula remains valid after its hypotheses, domain or convergence conditions are removed.

The inductive hypothesis may be used only after the base case and only to prove the next case.

Write the governing definition and list its hypotheses before calculating in mathematical induction.

Test a representative example and a boundary or counterexample, then verify the conclusion directly from the definition.