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Other Types of Induction

Learn about variations on induction. For example, if you can't topple the first domino, can you prove that toppling the 4th domino topples all of the ones after it?

Concept Quizzes

Induction - Problem Solving

         

Consider a statement \[S(N): 1+3+5+\cdots+(2N-1)=7+N^2,\] then which of the following is true?

What is the remainder when \({105}^{165}-1\) is divided by 4?

Consider a sequence \(\{a_n\}\) with \(a_1=5\) and \(a_2=13.\) If the sequence satisfies \[a_{n+2}=5a_{n+1}-6a_n\] for all positive integers \(n,\) what is \(a_{50}?\)

If \(\displaystyle a_n=2^{2^n}+1\) for \(n > 1,\) then what is the last digit of \(a_{451}?\)

Let \(P(n)\) be a statement involving a positive integer \(n.\) \(P(n+2)\) is true if \(P(n)\) or \(P(n+1)\) is true. Then what is the sufficient condition for the statement \(P(n)\) to be true for all positive integers?

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