Building the Next Case

Building the Next Case

IB Year 5 | Grade 11
Expose the previous case

The algebraic heart of induction is to rewrite the \(k+1\) case until the \(k\) case becomes visible. Only then may the induction hypothesis be substituted.

StructureStart the \(k+1\) case withWhere \(P(k)\) appears
Sum\(\displaystyle\sum_{r=1}^{k+1}u_r=\sum_{r=1}^{k}u_r+u_{k+1}\)The first \(k\) terms
RecurrenceWrite the given rule for \(u_{k+1}\).The occurrence of \(u_k\)
DivisibilityRewrite the new expression using the old expression plus a visible multiple of the divisor.A factor known to be divisible
Target-first workflow
  1. Write the exact target right-hand side for \(P(k+1)\).
  2. Start from the left-hand side of \(P(k+1)\), not from the desired right-hand side.
  3. Split off the new term.
  4. Substitute the induction hypothesis once the \(k\) case is visible.
  5. Factorise until the expression matches the target exactly.
Sum pattern

For a sum formula \(S_n=f(n)\), the target is \(S_{k+1}=f(k+1)\). After using \(S_{k+1}=S_k+u_{k+1}\) and the hypothesis \(S_k=f(k)\), all remaining work is ordinary algebra.

Exam check

Do not replace every \(k\) by \(k+1\) in the induction hypothesis. The hypothesis is known only at \(k\); the proof must build the next case.

Class check

If \(S_n=1+2+\cdots+n\), write the first two lines of the \(k+1\) case and identify the precise moment when \(S_k\) may be replaced.

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