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Problem Solving Skills

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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