Recurrence, Divisibility and Product Patterns

IB Year 5 | Grade 11

One logic, several algebra patterns

The logical skeleton of induction is unchanged across topics. What changes is the algebraic bridge that exposes the induction hypothesis.

Pattern	Useful hypothesis form	Typical next move
Recurrence	\(u_k=f(k)\)	Substitute into the recurrence for \(u_{k+1}\).
Divisibility by \(d\)	\(E_k=dm\) for some \(m\in\mathbb Z\)	Rewrite \(E_{k+1}\) as an integer multiple of \(d\).
Product identity	\(P_k=f(k)\)	Multiply by the new factor and cancel before expanding.
Telescoping expression	The closed form at \(k\)	Combine the old closed form with one new term.

Divisibility witness

For divisibility, “assume the expression is divisible by \(d\)” should immediately become “assume the expression equals \(dm\) for some integer \(m\)”. This creates the integer witness needed in the \(k+1\) case.

Product discipline

For products, cancellation is usually clearer than expansion. Factor each term first and preserve the product structure until the final target appears.

Index alignment

For recurrences, check the claimed formula uses the same starting index as the recurrence. An otherwise correct closed form can fail because of an index shift.

Pattern checks

Name every auxiliary integer introduced by a divisibility hypothesis.
Check that cancelled factors are non-zero on the stated domain.
Match the exponent or subscript of the target after the last line.
Keep the induction conclusion separate from the algebraic equality.

Class check

For \(7^n-1\), rewrite \(7^{k+1}-1\) so that the factor \(7^k-1\) appears without expanding powers.

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