Sequences and Series · Ages 16–17
Mathematical Induction: Proving Sum Formulas
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This worksheet is for 11th-grade students practicing mathematical induction proofs. Complete each task by writing down the required step or algebraic verification.
Test the base case n = 1 for the formula 1 + 3 + 5 + ... + (2n - 1) = n^2.
Write the induction hypothesis statement for the formula 1 + 2 + 3 + ... + n = n(n + 1)/2.
For the series 2 + 4 + 6 + ... + 2n = n(n + 1), what is the (k + 1)-th term that must be added to the induction hypothesis?
State the target algebraic expression for n = k + 1 when proving 1 + 3 + 5 + ... + (2n - 1) = n^2.
Given the hypothesis 1 + 3 + ... + (2k - 1) = k^2, add the (k + 1)-th term and simplify to prove the step for n = k + 1.
Complete the inductive step algebra for 2 + 4 + ... + 2k + 2(k + 1) using the hypothesis k(k + 1).
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Test the base case n = 1 for the formula 1 + 3 + 5 + ... + (2n - 1) = n^2.
Left side: 2(1) - 1 = 1. Right side: 1^2 = 1. Both sides equal 1, so the base case holds.
Write the induction hypothesis statement for the formula 1 + 2 + 3 + ... + n = n(n + 1)/2.
Assume the formula is true for n = k: 1 + 2 + ... + k = k(k + 1)/2 for some positive integer k.
For the series 2 + 4 + 6 + ... + 2n = n(n + 1), what is the (k + 1)-th term that must be added to the induction hypothesis?
The (k + 1)-th term is 2(k + 1), which simplifies to 2k + 2.
State the target algebraic expression for n = k + 1 when proving 1 + 3 + 5 + ... + (2n - 1) = n^2.
The target expression on the right side is (k + 1)^2, or k^2 + 2k + 1.
Given the hypothesis 1 + 3 + ... + (2k - 1) = k^2, add the (k + 1)-th term and simplify to prove the step for n = k + 1.
k^2 + (2(k + 1) - 1) = k^2 + 2k + 1 = (k + 1)^2. This matches the target formula.
Complete the inductive step algebra for 2 + 4 + ... + 2k + 2(k + 1) using the hypothesis k(k + 1).
k(k + 1) + 2(k + 1) = (k + 1)(k + 2). This matches the formula (k + 1)((k + 1) + 1).