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Mathematical induction: If n is an integer greater than or equal to 1, then the function nines ( n ) returns an integer made up
Mathematical induction:
If n is an integer greater than or equal to 1, then the function nines(n) returns an integer made up of n 9s. For example:
nines(1) | = | 9 |
nines(2) | = | 99 |
nines(3) | = | 999 |
nines(4) | = | 9999 |
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Let P(n) [nines(n) + 1 = 10n]. Prove that P(n) is true for all n 1, by mathematical induction. Use the induction schema [P(1) k [P(k) P(k + 1)]] n P(n).
1. (5 points.) What is the base case, P(1)?
2. (5 points.) What is the inductive case, k [P(k) P(k + 1)]?
3. (10 points.) Use your answers from questions 1 and 2 to construct the proof.
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