SME
Grade 12 Math · Problem type
The steps of a proof by induction
Use this for “Prove by induction that 1 + 3 + 5 + … + (2n − 1) = n² for every n ≥ 1.”
Put in order
Put the parts of the proof that 1 + 3 + … + (2n − 1) = n² in order.
1
?
2
?
3
?
4
?
5
?
Assumptions
•
Induction proves a statement for every whole number n ≥ 1, like a line of falling dominoes.
•
The base case knocks over the first domino; the step shows each one knocks over the next.
•
Both parts are needed: a step with no base case proves nothing.
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Related lessons
Mathematical induction
Main lesson
The sum of odd numbers
Use this for “Prove 1 + 3 + 5 + … + (2n − 1) = n² by induction.”
A geometric sum
Use this for “Prove 1 + 2 + 4 + … + 2ⁿ⁻¹ = 2ⁿ − 1 by induction” (r = 2), or any ratio r.
The sum of squares
Use this for “Prove 1² + 2² + … + n² = n(n + 1)(2n + 1)/6 by induction.”
A divisibility proof
Use this for “Prove that n³ − n is divisible by 3 for every whole number n ≥ 1.”