Proof by Induction (series) Worksheets
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Easy
10 problemsMedium
20 problemsHard
20 problemsMixed
30 problemsFree printable proof by induction (series) worksheets with step-by-step answer keys. Every worksheet is uniquely generated so students never see the same problems twice. Topics covered range from prove σi = n(n+1)/2 by induction at the easy level through to sum of squares / cubes at the advanced level.
What is proof by induction (series)?
Proof by induction for series is a method of proving formulas that describe the sum of sequences, such as showing that 1 + 2 + 3 + ... + n always equals n(n+1)/2. Students verify the formula works for a starting case, then prove that if it works for one value, it must work for the next, creating a domino effect that confirms the formula for all values.
This technique is essential in calculus and higher mathematics for establishing summation formulas and convergence properties. Engineers and computer scientists use induction to verify algorithms that process data in steps, and it appears in probability theory when analyzing expected values across multiple trials.
Pick a difficulty
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Beginner
Generate →- Concepts
- Prove Σi = n(n+1)/2 by induction
- Range
- arithmetic sum
- Steps
- 3 steps (basis, hypothesis, step)
- Example
- Prove 1+2+…+n = n(n+1)/2
Easy
Generate →- Concepts
- Arithmetic / odd-number sums
- Range
- Σi, Σ(2i−1)
- Steps
- 3 steps
- Example
- Prove 1+3+5+…+(2n−1) = n²
Medium
Generate →- Concepts
- Geometric-series closed form
- Range
- 1+r+…+rⁿ = (rⁿ⁺¹−1)/(r−1)
- Steps
- 3 steps
- Example
- Prove 1+4+…+4ⁿ = (4ⁿ⁺¹−1)/3
Hard
Generate →- Concepts
- Sum of squares / cubes
- Range
- Σi² , Σi³
- Steps
- 3 steps
- Example
- Prove Σi² = n(n+1)(2n+1)/6
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